mathlib3 documentation

topology.order.basic

Theory of topology on ordered spaces #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

Main definitions #

The order topology on an ordered space is the topology generated by all open intervals (or equivalently by those of the form (-∞, a) and (b, +∞)). We define it as preorder.topology α. However, we do not register it as an instance (as many existing ordered types already have topologies, which would be equal but not definitionally equal to preorder.topology α). Instead, we introduce a class order_topology α (which is a Prop, also known as a mixin) saying that on the type α having already a topological space structure and a preorder structure, the topological structure is equal to the order topology.

We also introduce another (mixin) class order_closed_topology α saying that the set of points (x, y) with x ≤ y is closed in the product space. This is automatically satisfied on a linear order with the order topology.

We prove many basic properties of such topologies.

Main statements #

This file contains the proofs of the following facts. For exact requirements (order_closed_topology vs order_topology, preorder vs partial_order vs linear_order etc) see their statements.

Open / closed sets #

is_open_lt : if f and g are continuous functions, then {x | f x < g x} is open;
is_open_Iio, is_open_Ioi, is_open_Ioo : open intervals are open;
is_closed_le : if f and g are continuous functions, then {x | f x ≤ g x} is closed;
is_closed_Iic, is_closed_Ici, is_closed_Icc : closed intervals are closed;
frontier_le_subset_eq, frontier_lt_subset_eq : frontiers of both {x | f x ≤ g x} and {x | f x < g x} are included by {x | f x = g x};
exists_Ioc_subset_of_mem_nhds, exists_Ico_subset_of_mem_nhds : if x < y, then any neighborhood of x includes an interval [x, z) for some z ∈ (x, y], and any neighborhood of y includes an interval (z, y] for some z ∈ [x, y).

Convergence and inequalities #

le_of_tendsto_of_tendsto : if f converges to a, g converges to b, and eventually f x ≤ g x, then a ≤ b
le_of_tendsto, ge_of_tendsto : if f converges to a and eventually f x ≤ b (resp., b ≤ f x), then a ≤ b (resp., b ≤ a); we also provide primed versions that assume the inequalities to hold for allx`.

Min, max, `Sup` and `Inf` #

continuous.min, continuous.max: pointwise min/max of two continuous functions is continuous.
tendsto.min, tendsto.max : if f tends to a and g tends to b, then their pointwise min/max tend to min a b and max a b, respectively.
tendsto_of_tendsto_of_tendsto_of_le_of_le : theorem known as squeeze theorem, sandwich theorem, theorem of Carabinieri, and two policemen (and a drunk) theorem; if g and h both converge to a, and eventually g x ≤ f x ≤ h x, then f converges to a.

Implementation notes #

We do not register the order topology as an instance on a preorder (or even on a linear order). Indeed, on many such spaces, a topology has already been constructed in a different way (think of the discrete spaces ℕ or ℤ, or ℝ that could inherit a topology as the completion of ℚ), and is in general not defeq to the one generated by the intervals. We make it available as a definition preorder.topology α though, that can be registered as an instance when necessary, or for specific types.

@[class]

structure order_closed_topology (α : Type u_1) [topological_space α] [preorder α] :

Prop

is_closed_le' : is_closed {p : α × α | p.fst ≤ p.snd}

A topology on a set which is both a topological space and a preorder is order-closed if the set of points (x, y) with x ≤ y is closed in the product space. We introduce this as a mixin. This property is satisfied for the order topology on a linear order, but it can be satisfied more generally, and suffices to derive many interesting properties relating order and topology.

Instances of this typeclass

@[protected, instance]

def order_dual.topological_space.first_countable_topology {α : Type u} [topological_space α] [h : topological_space.first_countable_topology α] :

topological_space.first_countable_topology αᵒᵈ

@[protected, instance]

def order_dual.topological_space.second_countable_topology {α : Type u} [topological_space α] [h : topological_space.second_countable_topology α] :

topological_space.second_countable_topology αᵒᵈ

theorem dense.order_dual {α : Type u} [topological_space α] {s : set α} (hs : dense s) :

dense (⇑order_dual.of_dual ⁻¹' s)

@[protected, instance]

def subtype.order_closed_topology {α : Type u} [topological_space α] [preorder α] [t : order_closed_topology α] {p : α → Prop} :

order_closed_topology (subtype p)

theorem is_closed_le_prod {α : Type u} [topological_space α] [preorder α] [t : order_closed_topology α] :

is_closed {p : α × α | p.fst ≤ p.snd}

theorem is_closed_le {α : Type u} {β : Type v} [topological_space α] [preorder α] [t : order_closed_topology α] [topological_space β] {f g : β → α} (hf : continuous f) (hg : continuous g) :

is_closed {b : β | f b ≤ g b}

theorem is_closed_le' {α : Type u} [topological_space α] [preorder α] [t : order_closed_topology α] (a : α) :

is_closed {b : α | b ≤ a}

theorem is_closed_Iic {α : Type u} [topological_space α] [preorder α] [t : order_closed_topology α] {a : α} :

is_closed (set.Iic a)

theorem is_closed_ge' {α : Type u} [topological_space α] [preorder α] [t : order_closed_topology α] (a : α) :

is_closed {b : α | a ≤ b}

theorem is_closed_Ici {α : Type u} [topological_space α] [preorder α] [t : order_closed_topology α] {a : α} :

is_closed (set.Ici a)

@[protected, instance]

def order_dual.order_closed_topology {α : Type u} [topological_space α] [preorder α] [t : order_closed_topology α] :

order_closed_topology αᵒᵈ

theorem is_closed_Icc {α : Type u} [topological_space α] [preorder α] [t : order_closed_topology α] {a b : α} :

is_closed (set.Icc a b)

@[simp]

theorem closure_Icc {α : Type u} [topological_space α] [preorder α] [t : order_closed_topology α] (a b : α) :

closure (set.Icc a b) = set.Icc a b

@[simp]

theorem closure_Iic {α : Type u} [topological_space α] [preorder α] [t : order_closed_topology α] (a : α) :

closure (set.Iic a) = set.Iic a

@[simp]

theorem closure_Ici {α : Type u} [topological_space α] [preorder α] [t : order_closed_topology α] (a : α) :

closure (set.Ici a) = set.Ici a

theorem le_of_tendsto_of_tendsto {α : Type u} {β : Type v} [topological_space α] [preorder α] [t : order_closed_topology α] {f g : β → α} {b : filter β} {a₁ a₂ : α} [b.ne_bot] (hf : filter.tendsto f b (nhds a₁)) (hg : filter.tendsto g b (nhds a₂)) (h : f ≤ᶠ[b] g) :

a₁ ≤ a₂

theorem tendsto_le_of_eventually_le {α : Type u} {β : Type v} [topological_space α] [preorder α] [t : order_closed_topology α] {f g : β → α} {b : filter β} {a₁ a₂ : α} [b.ne_bot] (hf : filter.tendsto f b (nhds a₁)) (hg : filter.tendsto g b (nhds a₂)) (h : f ≤ᶠ[b] g) :

a₁ ≤ a₂

Alias of le_of_tendsto_of_tendsto.

theorem le_of_tendsto_of_tendsto' {α : Type u} {β : Type v} [topological_space α] [preorder α] [t : order_closed_topology α] {f g : β → α} {b : filter β} {a₁ a₂ : α} [b.ne_bot] (hf : filter.tendsto f b (nhds a₁)) (hg : filter.tendsto g b (nhds a₂)) (h : ∀ (x : β), f x ≤ g x) :

a₁ ≤ a₂

theorem le_of_tendsto {α : Type u} {β : Type v} [topological_space α] [preorder α] [t : order_closed_topology α] {f : β → α} {a b : α} {x : filter β} [x.ne_bot] (lim : filter.tendsto f x (nhds a)) (h : ∀ᶠ (c : β) in x, f c ≤ b) :

a ≤ b

theorem le_of_tendsto' {α : Type u} {β : Type v} [topological_space α] [preorder α] [t : order_closed_topology α] {f : β → α} {a b : α} {x : filter β} [x.ne_bot] (lim : filter.tendsto f x (nhds a)) (h : ∀ (c : β), f c ≤ b) :

a ≤ b

theorem ge_of_tendsto {α : Type u} {β : Type v} [topological_space α] [preorder α] [t : order_closed_topology α] {f : β → α} {a b : α} {x : filter β} [x.ne_bot] (lim : filter.tendsto f x (nhds a)) (h : ∀ᶠ (c : β) in x, b ≤ f c) :

b ≤ a

theorem ge_of_tendsto' {α : Type u} {β : Type v} [topological_space α] [preorder α] [t : order_closed_topology α] {f : β → α} {a b : α} {x : filter β} [x.ne_bot] (lim : filter.tendsto f x (nhds a)) (h : ∀ (c : β), b ≤ f c) :

b ≤ a

@[simp]

theorem closure_le_eq {α : Type u} {β : Type v} [topological_space α] [preorder α] [t : order_closed_topology α] [topological_space β] {f g : β → α} (hf : continuous f) (hg : continuous g) :

closure {b : β | f b ≤ g b} = {b : β | f b ≤ g b}

theorem closure_lt_subset_le {α : Type u} {β : Type v} [topological_space α] [preorder α] [t : order_closed_topology α] [topological_space β] {f g : β → α} (hf : continuous f) (hg : continuous g) :

closure {b : β | f b < g b} ⊆ {b : β | f b ≤ g b}

theorem continuous_within_at.closure_le {α : Type u} {β : Type v} [topological_space α] [preorder α] [t : order_closed_topology α] [topological_space β] {f g : β → α} {s : set β} {x : β} (hx : x ∈ closure s) (hf : continuous_within_at f s x) (hg : continuous_within_at g s x) (h : ∀ (y : β), y ∈ s → f y ≤ g y) :

f x ≤ g x

theorem is_closed.is_closed_le {α : Type u} {β : Type v} [topological_space α] [preorder α] [t : order_closed_topology α] [topological_space β] {f g : β → α} {s : set β} (hs : is_closed s) (hf : continuous_on f s) (hg : continuous_on g s) :

is_closed {x ∈ s | f x ≤ g x}

If s is a closed set and two functions f and g are continuous on s, then the set {x ∈ s | f x ≤ g x} is a closed set.

theorem le_on_closure {α : Type u} {β : Type v} [topological_space α] [preorder α] [t : order_closed_topology α] [topological_space β] {f g : β → α} {s : set β} (h : ∀ (x : β), x ∈ s → f x ≤ g x) (hf : continuous_on f (closure s)) (hg : continuous_on g (closure s)) ⦃x : β⦄ (hx : x ∈ closure s) :

f x ≤ g x

theorem is_closed.epigraph {α : Type u} {β : Type v} [topological_space α] [preorder α] [t : order_closed_topology α] [topological_space β] {f : β → α} {s : set β} (hs : is_closed s) (hf : continuous_on f s) :

is_closed {p : β × α | p.fst ∈ s ∧ f p.fst ≤ p.snd}

theorem is_closed.hypograph {α : Type u} {β : Type v} [topological_space α] [preorder α] [t : order_closed_topology α] [topological_space β] {f : β → α} {s : set β} (hs : is_closed s) (hf : continuous_on f s) :

is_closed {p : β × α | p.fst ∈ s ∧ p.snd ≤ f p.fst}

theorem nhds_within_Ici_ne_bot {α : Type u} [topological_space α] [preorder α] {a b : α} (H₂ : a ≤ b) :

(nhds_within b (set.Ici a)).ne_bot

@[instance]

theorem nhds_within_Ici_self_ne_bot {α : Type u} [topological_space α] [preorder α] (a : α) :

(nhds_within a (set.Ici a)).ne_bot

theorem nhds_within_Iic_ne_bot {α : Type u} [topological_space α] [preorder α] {a b : α} (H : a ≤ b) :

(nhds_within a (set.Iic b)).ne_bot

@[instance]

theorem nhds_within_Iic_self_ne_bot {α : Type u} [topological_space α] [preorder α] (a : α) :

(nhds_within a (set.Iic a)).ne_bot

@[protected, instance]

def order_closed_topology.to_t2_space {α : Type u} [topological_space α] [partial_order α] [t : order_closed_topology α] :

theorem is_open_lt_prod {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] :

is_open {p : α × α | p.fst < p.snd}

theorem is_open_lt {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_closed_topology α] [topological_space β] {f g : β → α} (hf : continuous f) (hg : continuous g) :

is_open {b : β | f b < g b}

theorem is_open_Iio {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a : α} :

is_open (set.Iio a)

theorem is_open_Ioi {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a : α} :

is_open (set.Ioi a)

theorem is_open_Ioo {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b : α} :

is_open (set.Ioo a b)

@[simp]

theorem interior_Ioi {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a : α} :

interior (set.Ioi a) = set.Ioi a

@[simp]

theorem interior_Iio {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a : α} :

interior (set.Iio a) = set.Iio a

@[simp]

theorem interior_Ioo {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b : α} :

interior (set.Ioo a b) = set.Ioo a b

theorem Ioo_subset_closure_interior {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b : α} :

set.Ioo a b ⊆ closure (interior (set.Ioo a b))

theorem Iio_mem_nhds {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b : α} (h : a < b) :

set.Iio b ∈ nhds a

theorem Ioi_mem_nhds {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b : α} (h : a < b) :

set.Ioi a ∈ nhds b

theorem Iic_mem_nhds {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b : α} (h : a < b) :

set.Iic b ∈ nhds a

theorem Ici_mem_nhds {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b : α} (h : a < b) :

set.Ici a ∈ nhds b

theorem Ioo_mem_nhds {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b x : α} (ha : a < x) (hb : x < b) :

set.Ioo a b ∈ nhds x

theorem Ioc_mem_nhds {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b x : α} (ha : a < x) (hb : x < b) :

set.Ioc a b ∈ nhds x

theorem Ico_mem_nhds {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b x : α} (ha : a < x) (hb : x < b) :

set.Ico a b ∈ nhds x

theorem Icc_mem_nhds {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b x : α} (ha : a < x) (hb : x < b) :

set.Icc a b ∈ nhds x

theorem eventually_lt_of_tendsto_lt {α : Type u} {γ : Type w} [topological_space α] [linear_order α] [order_closed_topology α] {l : filter γ} {f : γ → α} {u v : α} (hv : v < u) (h : filter.tendsto f l (nhds v)) :

∀ᶠ (a : γ) in l, f a < u

theorem eventually_gt_of_tendsto_gt {α : Type u} {γ : Type w} [topological_space α] [linear_order α] [order_closed_topology α] {l : filter γ} {f : γ → α} {u v : α} (hv : u < v) (h : filter.tendsto f l (nhds v)) :

∀ᶠ (a : γ) in l, u < f a

theorem eventually_le_of_tendsto_lt {α : Type u} {γ : Type w} [topological_space α] [linear_order α] [order_closed_topology α] {l : filter γ} {f : γ → α} {u v : α} (hv : v < u) (h : filter.tendsto f l (nhds v)) :

∀ᶠ (a : γ) in l, f a ≤ u

theorem eventually_ge_of_tendsto_gt {α : Type u} {γ : Type w} [topological_space α] [linear_order α] [order_closed_topology α] {l : filter γ} {f : γ → α} {u v : α} (hv : u < v) (h : filter.tendsto f l (nhds v)) :

∀ᶠ (a : γ) in l, u ≤ f a

Neighborhoods to the left and to the right on an `order_closed_topology` #

Limits to the left and to the right of real functions are defined in terms of neighborhoods to the left and to the right, either open or closed, i.e., members of 𝓝[>] a and 𝓝[≥] a on the right, and similarly on the left. Here we simply prove that all right-neighborhoods of a point are equal, and we'll prove later other useful characterizations which require the stronger hypothesis order_topology α

Right neighborhoods, point excluded #

theorem Ioo_mem_nhds_within_Ioi {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b c : α} (H : b ∈ set.Ico a c) :

set.Ioo a c ∈ nhds_within b (set.Ioi b)

theorem Ioc_mem_nhds_within_Ioi {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b c : α} (H : b ∈ set.Ico a c) :

set.Ioc a c ∈ nhds_within b (set.Ioi b)

theorem Ico_mem_nhds_within_Ioi {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b c : α} (H : b ∈ set.Ico a c) :

set.Ico a c ∈ nhds_within b (set.Ioi b)

theorem Icc_mem_nhds_within_Ioi {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b c : α} (H : b ∈ set.Ico a c) :

set.Icc a c ∈ nhds_within b (set.Ioi b)

@[simp]

theorem nhds_within_Ioc_eq_nhds_within_Ioi {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b : α} (h : a < b) :

nhds_within a (set.Ioc a b) = nhds_within a (set.Ioi a)

@[simp]

theorem nhds_within_Ioo_eq_nhds_within_Ioi {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b : α} (h : a < b) :

nhds_within a (set.Ioo a b) = nhds_within a (set.Ioi a)

@[simp]

theorem continuous_within_at_Ioc_iff_Ioi {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_closed_topology α] [topological_space β] {a b : α} {f : α → β} (h : a < b) :

continuous_within_at f (set.Ioc a b) a ↔ continuous_within_at f (set.Ioi a) a

@[simp]

theorem continuous_within_at_Ioo_iff_Ioi {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_closed_topology α] [topological_space β] {a b : α} {f : α → β} (h : a < b) :

continuous_within_at f (set.Ioo a b) a ↔ continuous_within_at f (set.Ioi a) a

Left neighborhoods, point excluded #

theorem Ioo_mem_nhds_within_Iio {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b c : α} (H : b ∈ set.Ioc a c) :

set.Ioo a c ∈ nhds_within b (set.Iio b)

theorem Ico_mem_nhds_within_Iio {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b c : α} (H : b ∈ set.Ioc a c) :

set.Ico a c ∈ nhds_within b (set.Iio b)

theorem Ioc_mem_nhds_within_Iio {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b c : α} (H : b ∈ set.Ioc a c) :

set.Ioc a c ∈ nhds_within b (set.Iio b)

theorem Icc_mem_nhds_within_Iio {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b c : α} (H : b ∈ set.Ioc a c) :

set.Icc a c ∈ nhds_within b (set.Iio b)

@[simp]

theorem nhds_within_Ico_eq_nhds_within_Iio {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b : α} (h : a < b) :

nhds_within b (set.Ico a b) = nhds_within b (set.Iio b)

@[simp]

theorem nhds_within_Ioo_eq_nhds_within_Iio {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b : α} (h : a < b) :

nhds_within b (set.Ioo a b) = nhds_within b (set.Iio b)

@[simp]

theorem continuous_within_at_Ico_iff_Iio {α : Type u} {γ : Type w} [topological_space α] [linear_order α] [order_closed_topology α] [topological_space γ] {a b : α} {f : α → γ} (h : a < b) :

continuous_within_at f (set.Ico a b) b ↔ continuous_within_at f (set.Iio b) b

@[simp]

theorem continuous_within_at_Ioo_iff_Iio {α : Type u} {γ : Type w} [topological_space α] [linear_order α] [order_closed_topology α] [topological_space γ] {a b : α} {f : α → γ} (h : a < b) :

continuous_within_at f (set.Ioo a b) b ↔ continuous_within_at f (set.Iio b) b

Right neighborhoods, point included #

theorem Ioo_mem_nhds_within_Ici {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b c : α} (H : b ∈ set.Ioo a c) :

set.Ioo a c ∈ nhds_within b (set.Ici b)

theorem Ioc_mem_nhds_within_Ici {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b c : α} (H : b ∈ set.Ioo a c) :

set.Ioc a c ∈ nhds_within b (set.Ici b)

theorem Ico_mem_nhds_within_Ici {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b c : α} (H : b ∈ set.Ico a c) :

set.Ico a c ∈ nhds_within b (set.Ici b)

theorem Icc_mem_nhds_within_Ici {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b c : α} (H : b ∈ set.Ico a c) :

set.Icc a c ∈ nhds_within b (set.Ici b)

@[simp]

theorem nhds_within_Icc_eq_nhds_within_Ici {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b : α} (h : a < b) :

nhds_within a (set.Icc a b) = nhds_within a (set.Ici a)

@[simp]

theorem nhds_within_Ico_eq_nhds_within_Ici {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b : α} (h : a < b) :

nhds_within a (set.Ico a b) = nhds_within a (set.Ici a)

@[simp]

theorem continuous_within_at_Icc_iff_Ici {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_closed_topology α] [topological_space β] {a b : α} {f : α → β} (h : a < b) :

continuous_within_at f (set.Icc a b) a ↔ continuous_within_at f (set.Ici a) a

@[simp]

theorem continuous_within_at_Ico_iff_Ici {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_closed_topology α] [topological_space β] {a b : α} {f : α → β} (h : a < b) :

continuous_within_at f (set.Ico a b) a ↔ continuous_within_at f (set.Ici a) a

Left neighborhoods, point included #

theorem Ioo_mem_nhds_within_Iic {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b c : α} (H : b ∈ set.Ioo a c) :

set.Ioo a c ∈ nhds_within b (set.Iic b)

theorem Ico_mem_nhds_within_Iic {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b c : α} (H : b ∈ set.Ioo a c) :

set.Ico a c ∈ nhds_within b (set.Iic b)

theorem Ioc_mem_nhds_within_Iic {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b c : α} (H : b ∈ set.Ioc a c) :

set.Ioc a c ∈ nhds_within b (set.Iic b)

theorem Icc_mem_nhds_within_Iic {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b c : α} (H : b ∈ set.Ioc a c) :

set.Icc a c ∈ nhds_within b (set.Iic b)

@[simp]

theorem nhds_within_Icc_eq_nhds_within_Iic {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b : α} (h : a < b) :

nhds_within b (set.Icc a b) = nhds_within b (set.Iic b)

@[simp]

theorem nhds_within_Ioc_eq_nhds_within_Iic {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b : α} (h : a < b) :

nhds_within b (set.Ioc a b) = nhds_within b (set.Iic b)

@[simp]

theorem continuous_within_at_Icc_iff_Iic {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_closed_topology α] [topological_space β] {a b : α} {f : α → β} (h : a < b) :

continuous_within_at f (set.Icc a b) b ↔ continuous_within_at f (set.Iic b) b

@[simp]

theorem continuous_within_at_Ioc_iff_Iic {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_closed_topology α] [topological_space β] {a b : α} {f : α → β} (h : a < b) :

continuous_within_at f (set.Ioc a b) b ↔ continuous_within_at f (set.Iic b) b

theorem lt_subset_interior_le {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_closed_topology α] {f g : β → α} [topological_space β] (hf : continuous f) (hg : continuous g) :

{b : β | f b < g b} ⊆ interior {b : β | f b ≤ g b}

theorem frontier_le_subset_eq {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_closed_topology α] {f g : β → α} [topological_space β] (hf : continuous f) (hg : continuous g) :

frontier {b : β | f b ≤ g b} ⊆ {b : β | f b = g b}

theorem frontier_Iic_subset {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] (a : α) :

frontier (set.Iic a) ⊆ {a}

theorem frontier_Ici_subset {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] (a : α) :

frontier (set.Ici a) ⊆ {a}

theorem frontier_lt_subset_eq {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_closed_topology α] {f g : β → α} [topological_space β] (hf : continuous f) (hg : continuous g) :

frontier {b : β | f b < g b} ⊆ {b : β | f b = g b}

theorem continuous_if_le {α : Type u} {β : Type v} {γ : Type w} [topological_space α] [linear_order α] [order_closed_topology α] {f g : β → α} [topological_space β] [topological_space γ] [Π (x : β), decidable (f x ≤ g x)] {f' g' : β → γ} (hf : continuous f) (hg : continuous g) (hf' : continuous_on f' {x : β | f x ≤ g x}) (hg' : continuous_on g' {x : β | g x ≤ f x}) (hfg : ∀ (x : β), f x = g x → f' x = g' x) :

continuous (λ (x : β), ite (f x ≤ g x) (f' x) (g' x))

theorem continuous.if_le {α : Type u} {β : Type v} {γ : Type w} [topological_space α] [linear_order α] [order_closed_topology α] {f g : β → α} [topological_space β] [topological_space γ] [Π (x : β), decidable (f x ≤ g x)] {f' g' : β → γ} (hf' : continuous f') (hg' : continuous g') (hf : continuous f) (hg : continuous g) (hfg : ∀ (x : β), f x = g x → f' x = g' x) :

continuous (λ (x : β), ite (f x ≤ g x) (f' x) (g' x))

theorem tendsto.eventually_lt {α : Type u} {γ : Type w} [topological_space α] [linear_order α] [order_closed_topology α] {l : filter γ} {f g : γ → α} {y z : α} (hf : filter.tendsto f l (nhds y)) (hg : filter.tendsto g l (nhds z)) (hyz : y < z) :

∀ᶠ (x : γ) in l, f x < g x

theorem continuous_at.eventually_lt {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_closed_topology α] {f g : β → α} [topological_space β] {x₀ : β} (hf : continuous_at f x₀) (hg : continuous_at g x₀) (hfg : f x₀ < g x₀) :

∀ᶠ (x : β) in nhds x₀, f x < g x

@[continuity]

theorem continuous.min {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_closed_topology α] {f g : β → α} [topological_space β] (hf : continuous f) (hg : continuous g) :

continuous (λ (b : β), linear_order.min (f b) (g b))

@[continuity]

theorem continuous.max {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_closed_topology α] {f g : β → α} [topological_space β] (hf : continuous f) (hg : continuous g) :

continuous (λ (b : β), linear_order.max (f b) (g b))

theorem continuous_min {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] :

continuous (λ (p : α × α), linear_order.min p.fst p.snd)

theorem continuous_max {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] :

continuous (λ (p : α × α), linear_order.max p.fst p.snd)

theorem filter.tendsto.max {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_closed_topology α] {f g : β → α} {b : filter β} {a₁ a₂ : α} (hf : filter.tendsto f b (nhds a₁)) (hg : filter.tendsto g b (nhds a₂)) :

filter.tendsto (λ (b : β), linear_order.max (f b) (g b)) b (nhds (linear_order.max a₁ a₂))

theorem filter.tendsto.min {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_closed_topology α] {f g : β → α} {b : filter β} {a₁ a₂ : α} (hf : filter.tendsto f b (nhds a₁)) (hg : filter.tendsto g b (nhds a₂)) :

filter.tendsto (λ (b : β), linear_order.min (f b) (g b)) b (nhds (linear_order.min a₁ a₂))

theorem filter.tendsto.max_right {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_closed_topology α] {f : β → α} {l : filter β} {a : α} (h : filter.tendsto f l (nhds a)) :

filter.tendsto (λ (i : β), linear_order.max a (f i)) l (nhds a)

theorem filter.tendsto.max_left {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_closed_topology α] {f : β → α} {l : filter β} {a : α} (h : filter.tendsto f l (nhds a)) :

filter.tendsto (λ (i : β), linear_order.max (f i) a) l (nhds a)

theorem filter.tendsto_nhds_max_right {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_closed_topology α] {f : β → α} {l : filter β} {a : α} (h : filter.tendsto f l (nhds_within a (set.Ioi a))) :

filter.tendsto (λ (i : β), linear_order.max a (f i)) l (nhds_within a (set.Ioi a))

theorem filter.tendsto_nhds_max_left {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_closed_topology α] {f : β → α} {l : filter β} {a : α} (h : filter.tendsto f l (nhds_within a (set.Ioi a))) :

filter.tendsto (λ (i : β), linear_order.max (f i) a) l (nhds_within a (set.Ioi a))

theorem filter.tendsto.min_right {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_closed_topology α] {f : β → α} {l : filter β} {a : α} (h : filter.tendsto f l (nhds a)) :

filter.tendsto (λ (i : β), linear_order.min a (f i)) l (nhds a)

theorem filter.tendsto.min_left {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_closed_topology α] {f : β → α} {l : filter β} {a : α} (h : filter.tendsto f l (nhds a)) :

filter.tendsto (λ (i : β), linear_order.min (f i) a) l (nhds a)

theorem filter.tendsto_nhds_min_right {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_closed_topology α] {f : β → α} {l : filter β} {a : α} (h : filter.tendsto f l (nhds_within a (set.Iio a))) :

filter.tendsto (λ (i : β), linear_order.min a (f i)) l (nhds_within a (set.Iio a))

theorem filter.tendsto_nhds_min_left {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_closed_topology α] {f : β → α} {l : filter β} {a : α} (h : filter.tendsto f l (nhds_within a (set.Iio a))) :

filter.tendsto (λ (i : β), linear_order.min (f i) a) l (nhds_within a (set.Iio a))

theorem dense.exists_lt {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] [no_min_order α] {s : set α} (hs : dense s) (x : α) :

∃ (y : α) (H : y ∈ s), y < x

theorem dense.exists_gt {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] [no_max_order α] {s : set α} (hs : dense s) (x : α) :

∃ (y : α) (H : y ∈ s), x < y

theorem dense.exists_le {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] [no_min_order α] {s : set α} (hs : dense s) (x : α) :

∃ (y : α) (H : y ∈ s), y ≤ x

theorem dense.exists_ge {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] [no_max_order α] {s : set α} (hs : dense s) (x : α) :

∃ (y : α) (H : y ∈ s), x ≤ y

theorem dense.exists_le' {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {s : set α} (hs : dense s) (hbot : ∀ (x : α), is_bot x → x ∈ s) (x : α) :

∃ (y : α) (H : y ∈ s), y ≤ x

theorem dense.exists_ge' {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {s : set α} (hs : dense s) (htop : ∀ (x : α), is_top x → x ∈ s) (x : α) :

∃ (y : α) (H : y ∈ s), x ≤ y

theorem dense.exists_between {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] [densely_ordered α] {s : set α} (hs : dense s) {x y : α} (h : x < y) :

∃ (z : α) (H : z ∈ s), z ∈ set.Ioo x y

@[protected, instance]

def prod.order_closed_topology {α : Type u} {β : Type v} [preorder α] [topological_space α] [order_closed_topology α] [preorder β] [topological_space β] [order_closed_topology β] :

order_closed_topology (α × β)

@[protected, instance]

def pi.order_closed_topology {ι : Type u_1} {α : ι → Type u_2} [Π (i : ι), preorder (α i)] [Π (i : ι), topological_space (α i)] [∀ (i : ι), order_closed_topology (α i)] :

order_closed_topology (Π (i : ι), α i)

@[protected, instance]

def pi.order_closed_topology' {α : Type u} {β : Type v} [preorder β] [topological_space β] [order_closed_topology β] :

order_closed_topology (α → β)

@[class]

structure order_topology (α : Type u_1) [t : topological_space α] [preorder α] :

Prop

topology_eq_generate_intervals : t = topological_space.generate_from {s : set α | ∃ (a : α), s = set.Ioi a ∨ s = set.Iio a}

The order topology on an ordered type is the topology generated by open intervals. We register it on a preorder, but it is mostly interesting in linear orders, where it is also order-closed. We define it as a mixin. If you want to introduce the order topology on a preorder, use preorder.topology.

Instances of this typeclass

def preorder.topology (α : Type u_1) [preorder α] :

topological_space α

(Order) topology on a partial order α generated by the subbase of open intervals (a, ∞) = { x ∣ a < x }, (-∞ , b) = {x ∣ x < b} for all a, b in α. We do not register it as an instance as many ordered sets are already endowed with the same topology, most often in a non-defeq way though. Register as a local instance when necessary.

Equations

preorder.topology α = topological_space.generate_from {s : set α | ∃ (a : α), s = {b : α | a < b} ∨ s = {b : α | b < a}}

@[protected, instance]

def order_dual.order_topology {α : Type u} [topological_space α] [preorder α] [t : order_topology α] :

order_topology αᵒᵈ

theorem is_open_iff_generate_intervals {α : Type u} [topological_space α] [preorder α] [t : order_topology α] {s : set α} :

is_open s ↔ topological_space.generate_open {s : set α | ∃ (a : α), s = set.Ioi a ∨ s = set.Iio a} s

theorem is_open_lt' {α : Type u} [topological_space α] [preorder α] [t : order_topology α] (a : α) :

is_open {b : α | a < b}

theorem is_open_gt' {α : Type u} [topological_space α] [preorder α] [t : order_topology α] (a : α) :

is_open {b : α | b < a}

theorem lt_mem_nhds {α : Type u} [topological_space α] [preorder α] [t : order_topology α] {a b : α} (h : a < b) :

∀ᶠ (x : α) in nhds b, a < x

theorem le_mem_nhds {α : Type u} [topological_space α] [preorder α] [t : order_topology α] {a b : α} (h : a < b) :

∀ᶠ (x : α) in nhds b, a ≤ x

theorem gt_mem_nhds {α : Type u} [topological_space α] [preorder α] [t : order_topology α] {a b : α} (h : a < b) :

∀ᶠ (x : α) in nhds a, x < b

theorem ge_mem_nhds {α : Type u} [topological_space α] [preorder α] [t : order_topology α] {a b : α} (h : a < b) :

∀ᶠ (x : α) in nhds a, x ≤ b

theorem nhds_eq_order {α : Type u} [topological_space α] [preorder α] [t : order_topology α] (a : α) :

nhds a = (⨅ (b : α) (H : b ∈ set.Iio a), filter.principal (set.Ioi b)) ⊓ ⨅ (b : α) (H : b ∈ set.Ioi a), filter.principal (set.Iio b)

theorem tendsto_order {α : Type u} {β : Type v} [topological_space α] [preorder α] [t : order_topology α] {f : β → α} {a : α} {x : filter β} :

filter.tendsto f x (nhds a) ↔ (∀ (a' : α), a' < a → (∀ᶠ (b : β) in x, a' < f b)) ∧ ∀ (a' : α), a' > a → (∀ᶠ (b : β) in x, f b < a')

@[protected, instance]

def tendsto_Icc_class_nhds {α : Type u} [topological_space α] [preorder α] [t : order_topology α] (a : α) :

filter.tendsto_Ixx_class set.Icc (nhds a) (nhds a)

@[protected, instance]

def tendsto_Ico_class_nhds {α : Type u} [topological_space α] [preorder α] [t : order_topology α] (a : α) :

filter.tendsto_Ixx_class set.Ico (nhds a) (nhds a)

@[protected, instance]

def tendsto_Ioc_class_nhds {α : Type u} [topological_space α] [preorder α] [t : order_topology α] (a : α) :

filter.tendsto_Ixx_class set.Ioc (nhds a) (nhds a)

@[protected, instance]

def tendsto_Ioo_class_nhds {α : Type u} [topological_space α] [preorder α] [t : order_topology α] (a : α) :

filter.tendsto_Ixx_class set.Ioo (nhds a) (nhds a)

theorem tendsto_of_tendsto_of_tendsto_of_le_of_le' {α : Type u} {β : Type v} [topological_space α] [preorder α] [t : order_topology α] {f g h : β → α} {b : filter β} {a : α} (hg : filter.tendsto g b (nhds a)) (hh : filter.tendsto h b (nhds a)) (hgf : ∀ᶠ (b : β) in b, g b ≤ f b) (hfh : ∀ᶠ (b : β) in b, f b ≤ h b) :

filter.tendsto f b (nhds a)

Squeeze theorem (also known as sandwich theorem). This version assumes that inequalities hold eventually for the filter.

theorem tendsto_of_tendsto_of_tendsto_of_le_of_le {α : Type u} {β : Type v} [topological_space α] [preorder α] [t : order_topology α] {f g h : β → α} {b : filter β} {a : α} (hg : filter.tendsto g b (nhds a)) (hh : filter.tendsto h b (nhds a)) (hgf : g ≤ f) (hfh : f ≤ h) :

filter.tendsto f b (nhds a)

Squeeze theorem (also known as sandwich theorem). This version assumes that inequalities hold everywhere.

theorem nhds_order_unbounded {α : Type u} [topological_space α] [preorder α] [t : order_topology α] {a : α} (hu : ∃ (u : α), a < u) (hl : ∃ (l : α), l < a) :

nhds a = ⨅ (l : α) (h₂ : l < a) (u : α) (h₂ : a < u), filter.principal (set.Ioo l u)

theorem tendsto_order_unbounded {α : Type u} {β : Type v} [topological_space α] [preorder α] [t : order_topology α] {f : β → α} {a : α} {x : filter β} (hu : ∃ (u : α), a < u) (hl : ∃ (l : α), l < a) (h : ∀ (l u : α), l < a → a < u → (∀ᶠ (b : β) in x, l < f b ∧ f b < u)) :

filter.tendsto f x (nhds a)

@[protected, instance]

def tendsto_Ixx_nhds_within {α : Type u_1} [preorder α] [topological_space α] (a : α) {s t : set α} {Ixx : α → α → set α} [filter.tendsto_Ixx_class Ixx (nhds a) (nhds a)] [filter.tendsto_Ixx_class Ixx (filter.principal s) (filter.principal t)] :

filter.tendsto_Ixx_class Ixx (nhds_within a s) (nhds_within a t)

@[protected, instance]

def tendsto_Icc_class_nhds_pi {ι : Type u_1} {α : ι → Type u_2} [Π (i : ι), preorder (α i)] [Π (i : ι), topological_space (α i)] [∀ (i : ι), order_topology (α i)] (f : Π (i : ι), α i) :

filter.tendsto_Ixx_class set.Icc (nhds f) (nhds f)

theorem induced_order_topology' {α : Type u} {β : Type v} [preorder α] [ta : topological_space β] [preorder β] [order_topology β] (f : α → β) (hf : ∀ {x y : α}, f x < f y ↔ x < y) (H₁ : ∀ {a : α} {x : β}, x < f a → (∃ (b : α) (H : b < a), x ≤ f b)) (H₂ : ∀ {a : α} {x : β}, f a < x → (∃ (b : α) (H : b > a), f b ≤ x)) :

order_topology α

theorem induced_order_topology {α : Type u} {β : Type v} [preorder α] [ta : topological_space β] [preorder β] [order_topology β] (f : α → β) (hf : ∀ {x y : α}, f x < f y ↔ x < y) (H : ∀ {x y : β}, x < y → (∃ (a : α), x < f a ∧ f a < y)) :

order_topology α

@[protected, instance]

def order_topology_of_ord_connected {α : Type u} [ta : topological_space α] [linear_order α] [order_topology α] {t : set α} [ht : t.ord_connected] :

order_topology ↥t

On an ord_connected subset of a linear order, the order topology for the restriction of the order is the same as the restriction to the subset of the order topology.

theorem nhds_within_Ici_eq'' {α : Type u} [topological_space α] [preorder α] [order_topology α] (a : α) :

nhds_within a (set.Ici a) = (⨅ (u : α) (hu : a < u), filter.principal (set.Iio u)) ⊓ filter.principal (set.Ici a)

theorem nhds_within_Iic_eq'' {α : Type u} [topological_space α] [preorder α] [order_topology α] (a : α) :

nhds_within a (set.Iic a) = (⨅ (l : α) (H : l < a), filter.principal (set.Ioi l)) ⊓ filter.principal (set.Iic a)

theorem nhds_within_Ici_eq' {α : Type u} [topological_space α] [preorder α] [order_topology α] {a : α} (ha : ∃ (u : α), a < u) :

nhds_within a (set.Ici a) = ⨅ (u : α) (hu : a < u), filter.principal (set.Ico a u)

theorem nhds_within_Iic_eq' {α : Type u} [topological_space α] [preorder α] [order_topology α] {a : α} (ha : ∃ (l : α), l < a) :

nhds_within a (set.Iic a) = ⨅ (l : α) (H : l < a), filter.principal (set.Ioc l a)

theorem nhds_within_Ici_basis' {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a : α} (ha : ∃ (u : α), a < u) :

(nhds_within a (set.Ici a)).has_basis (λ (u : α), a < u) (λ (u : α), set.Ico a u)

theorem nhds_within_Iic_basis' {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a : α} (ha : ∃ (l : α), l < a) :

(nhds_within a (set.Iic a)).has_basis (λ (l : α), l < a) (λ (l : α), set.Ioc l a)

theorem nhds_within_Ici_basis {α : Type u} [topological_space α] [linear_order α] [order_topology α] [no_max_order α] (a : α) :

(nhds_within a (set.Ici a)).has_basis (λ (u : α), a < u) (λ (u : α), set.Ico a u)

theorem nhds_within_Iic_basis {α : Type u} [topological_space α] [linear_order α] [order_topology α] [no_min_order α] (a : α) :

(nhds_within a (set.Iic a)).has_basis (λ (l : α), l < a) (λ (l : α), set.Ioc l a)

theorem nhds_top_order {α : Type u} [topological_space α] [preorder α] [order_top α] [order_topology α] :

nhds ⊤ = ⨅ (l : α) (h₂ : l < ⊤), filter.principal (set.Ioi l)

theorem nhds_bot_order {α : Type u} [topological_space α] [preorder α] [order_bot α] [order_topology α] :

nhds ⊥ = ⨅ (l : α) (h₂ : ⊥ < l), filter.principal (set.Iio l)

theorem nhds_top_basis {α : Type u} [topological_space α] [linear_order α] [order_top α] [order_topology α] [nontrivial α] :

(nhds ⊤).has_basis (λ (a : α), a < ⊤) (λ (a : α), set.Ioi a)

theorem nhds_bot_basis {α : Type u} [topological_space α] [linear_order α] [order_bot α] [order_topology α] [nontrivial α] :

(nhds ⊥).has_basis (λ (a : α), ⊥ < a) (λ (a : α), set.Iio a)

theorem nhds_top_basis_Ici {α : Type u} [topological_space α] [linear_order α] [order_top α] [order_topology α] [nontrivial α] [densely_ordered α] :

(nhds ⊤).has_basis (λ (a : α), a < ⊤) set.Ici

theorem nhds_bot_basis_Iic {α : Type u} [topological_space α] [linear_order α] [order_bot α] [order_topology α] [nontrivial α] [densely_ordered α] :

(nhds ⊥).has_basis (λ (a : α), ⊥ < a) set.Iic

theorem tendsto_nhds_top_mono {α : Type u} {β : Type v} [topological_space β] [preorder β] [order_top β] [order_topology β] {l : filter α} {f g : α → β} (hf : filter.tendsto f l (nhds ⊤)) (hg : f ≤ᶠ[l] g) :

filter.tendsto g l (nhds ⊤)

theorem tendsto_nhds_bot_mono {α : Type u} {β : Type v} [topological_space β] [preorder β] [order_bot β] [order_topology β] {l : filter α} {f g : α → β} (hf : filter.tendsto f l (nhds ⊥)) (hg : g ≤ᶠ[l] f) :

filter.tendsto g l (nhds ⊥)

theorem tendsto_nhds_top_mono' {α : Type u} {β : Type v} [topological_space β] [preorder β] [order_top β] [order_topology β] {l : filter α} {f g : α → β} (hf : filter.tendsto f l (nhds ⊤)) (hg : f ≤ g) :

filter.tendsto g l (nhds ⊤)

theorem tendsto_nhds_bot_mono' {α : Type u} {β : Type v} [topological_space β] [preorder β] [order_bot β] [order_topology β] {l : filter α} {f g : α → β} (hf : filter.tendsto f l (nhds ⊥)) (hg : g ≤ f) :

filter.tendsto g l (nhds ⊥)

theorem eventually_le_nhds {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b : α} (hab : a < b) :

∀ᶠ (x : α) in nhds a, x ≤ b

theorem eventually_lt_nhds {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b : α} (hab : a < b) :

∀ᶠ (x : α) in nhds a, x < b

theorem eventually_ge_nhds {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b : α} (hab : b < a) :

∀ᶠ (x : α) in nhds a, b ≤ x

theorem eventually_gt_nhds {α : Type u} [topological_space α] [linear_order α] [order_closed_topology α] {a b : α} (hab : b < a) :

∀ᶠ (x : α) in nhds a, b < x

theorem order_separated {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a₁ a₂ : α} (h : a₁ < a₂) :

∃ (u v : set α), is_open u ∧ is_open v ∧ a₁ ∈ u ∧ a₂ ∈ v ∧ ∀ (b₁ : α), b₁ ∈ u → ∀ (b₂ : α), b₂ ∈ v → b₁ < b₂

@[protected, instance]

def order_topology.to_order_closed_topology {α : Type u} [topological_space α] [linear_order α] [order_topology α] :

order_closed_topology α

theorem exists_Ioc_subset_of_mem_nhds {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a : α} {s : set α} (hs : s ∈ nhds a) (h : ∃ (l : α), l < a) :

∃ (l : α) (H : l < a), set.Ioc l a ⊆ s

theorem exists_Ioc_subset_of_mem_nhds' {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a : α} {s : set α} (hs : s ∈ nhds a) {l : α} (hl : l < a) :

∃ (l' : α) (H : l' ∈ set.Ico l a), set.Ioc l' a ⊆ s

theorem exists_Ico_subset_of_mem_nhds' {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a : α} {s : set α} (hs : s ∈ nhds a) {u : α} (hu : a < u) :

∃ (u' : α) (H : u' ∈ set.Ioc a u), set.Ico a u' ⊆ s

theorem exists_Ico_subset_of_mem_nhds {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a : α} {s : set α} (hs : s ∈ nhds a) (h : ∃ (u : α), a < u) :

∃ (u : α) (_x : a < u), set.Ico a u ⊆ s

theorem exists_Icc_mem_subset_of_mem_nhds_within_Ici {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a : α} {s : set α} (hs : s ∈ nhds_within a (set.Ici a)) :

∃ (b : α) (_x : a ≤ b), set.Icc a b ∈ nhds_within a (set.Ici a) ∧ set.Icc a b ⊆ s

theorem exists_Icc_mem_subset_of_mem_nhds_within_Iic {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a : α} {s : set α} (hs : s ∈ nhds_within a (set.Iic a)) :

∃ (b : α) (H : b ≤ a), set.Icc b a ∈ nhds_within a (set.Iic a) ∧ set.Icc b a ⊆ s

theorem exists_Icc_mem_subset_of_mem_nhds {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a : α} {s : set α} (hs : s ∈ nhds a) :

∃ (b c : α), a ∈ set.Icc b c ∧ set.Icc b c ∈ nhds a ∧ set.Icc b c ⊆ s

theorem is_open.exists_Ioo_subset {α : Type u} [topological_space α] [linear_order α] [order_topology α] [nontrivial α] {s : set α} (hs : is_open s) (h : s.nonempty) :

∃ (a b : α), a < b ∧ set.Ioo a b ⊆ s

theorem dense_of_exists_between {α : Type u} [topological_space α] [linear_order α] [order_topology α] [nontrivial α] {s : set α} (h : ∀ ⦃a b : α⦄, a < b → (∃ (c : α) (H : c ∈ s), a < c ∧ c < b)) :

theorem dense_iff_exists_between {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] [nontrivial α] {s : set α} :

dense s ↔ ∀ (a b : α), a < b → (∃ (c : α) (H : c ∈ s), a < c ∧ c < b)

A set in a nontrivial densely linear ordered type is dense in the sense of topology if and only if for any a < b there exists c ∈ s, a < c < b. Each implication requires less typeclass assumptions.

theorem mem_nhds_iff_exists_Ioo_subset' {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a : α} {s : set α} (hl : ∃ (l : α), l < a) (hu : ∃ (u : α), a < u) :

s ∈ nhds a ↔ ∃ (l u : α), a ∈ set.Ioo l u ∧ set.Ioo l u ⊆ s

A set is a neighborhood of a if and only if it contains an interval (l, u) containing a, provided a is neither a bottom element nor a top element.

theorem mem_nhds_iff_exists_Ioo_subset {α : Type u} [topological_space α] [linear_order α] [order_topology α] [no_max_order α] [no_min_order α] {a : α} {s : set α} :

s ∈ nhds a ↔ ∃ (l u : α), a ∈ set.Ioo l u ∧ set.Ioo l u ⊆ s

A set is a neighborhood of a if and only if it contains an interval (l, u) containing a.

theorem nhds_basis_Ioo' {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a : α} (hl : ∃ (l : α), l < a) (hu : ∃ (u : α), a < u) :

(nhds a).has_basis (λ (b : α × α), b.fst < a ∧ a < b.snd) (λ (b : α × α), set.Ioo b.fst b.snd)

theorem nhds_basis_Ioo {α : Type u} [topological_space α] [linear_order α] [order_topology α] [no_max_order α] [no_min_order α] (a : α) :

(nhds a).has_basis (λ (b : α × α), b.fst < a ∧ a < b.snd) (λ (b : α × α), set.Ioo b.fst b.snd)

theorem filter.eventually.exists_Ioo_subset {α : Type u} [topological_space α] [linear_order α] [order_topology α] [no_max_order α] [no_min_order α] {a : α} {p : α → Prop} (hp : ∀ᶠ (x : α) in nhds a, p x) :

∃ (l u : α), a ∈ set.Ioo l u ∧ set.Ioo l u ⊆ {x : α | p x}

theorem countable_of_isolated_right {α : Type u} [topological_space α] [linear_order α] [order_topology α] [topological_space.second_countable_topology α] :

{x : α | ∃ (y : α), x < y ∧ set.Ioo x y = ∅}.countable

The set of points which are isolated on the right is countable when the space is second-countable.

theorem countable_of_isolated_left {α : Type u} [topological_space α] [linear_order α] [order_topology α] [topological_space.second_countable_topology α] :

{x : α | ∃ (y : α), y < x ∧ set.Ioo y x = ∅}.countable

The set of points which are isolated on the left is countable when the space is second-countable.

theorem set.pairwise_disjoint.countable_of_Ioo {α : Type u} [topological_space α] [linear_order α] [order_topology α] [topological_space.second_countable_topology α] {y : α → α} {s : set α} (h : s.pairwise_disjoint (λ (x : α), set.Ioo x (y x))) (h' : ∀ (x : α), x ∈ s → x < y x) :

Consider a disjoint family of intervals (x, y) with x < y in a second-countable space. Then the family is countable. This is not a straightforward consequence of second-countability as some of these intervals might be empty (but in fact this can happen only for countably many of them).

Intervals in `Π i, π i` belong to `𝓝 x` #

For each lemma pi_Ixx_mem_nhds we add a non-dependent version pi_Ixx_mem_nhds' because sometimes Lean fails to unify different instances while trying to apply the dependent version to, e.g., ι → ℝ.

theorem pi_Iic_mem_nhds {ι : Type u_1} {π : ι → Type u_2} [finite ι] [Π (i : ι), linear_order (π i)] [Π (i : ι), topological_space (π i)] [∀ (i : ι), order_topology (π i)] {a x : Π (i : ι), π i} (ha : ∀ (i : ι), x i < a i) :

set.Iic a ∈ nhds x

theorem pi_Iic_mem_nhds' {α : Type u} [topological_space α] [linear_order α] [order_topology α] {ι : Type u_1} [finite ι] {a' x' : ι → α} (ha : ∀ (i : ι), x' i < a' i) :

set.Iic a' ∈ nhds x'

theorem pi_Ici_mem_nhds {ι : Type u_1} {π : ι → Type u_2} [finite ι] [Π (i : ι), linear_order (π i)] [Π (i : ι), topological_space (π i)] [∀ (i : ι), order_topology (π i)] {a x : Π (i : ι), π i} (ha : ∀ (i : ι), a i < x i) :

set.Ici a ∈ nhds x

theorem pi_Ici_mem_nhds' {α : Type u} [topological_space α] [linear_order α] [order_topology α] {ι : Type u_1} [finite ι] {a' x' : ι → α} (ha : ∀ (i : ι), a' i < x' i) :

set.Ici a' ∈ nhds x'

theorem pi_Icc_mem_nhds {ι : Type u_1} {π : ι → Type u_2} [finite ι] [Π (i : ι), linear_order (π i)] [Π (i : ι), topological_space (π i)] [∀ (i : ι), order_topology (π i)] {a b x : Π (i : ι), π i} (ha : ∀ (i : ι), a i < x i) (hb : ∀ (i : ι), x i < b i) :

set.Icc a b ∈ nhds x

theorem pi_Icc_mem_nhds' {α : Type u} [topological_space α] [linear_order α] [order_topology α] {ι : Type u_1} [finite ι] {a' b' x' : ι → α} (ha : ∀ (i : ι), a' i < x' i) (hb : ∀ (i : ι), x' i < b' i) :

set.Icc a' b' ∈ nhds x'

theorem pi_Iio_mem_nhds {ι : Type u_1} {π : ι → Type u_2} [finite ι] [Π (i : ι), linear_order (π i)] [Π (i : ι), topological_space (π i)] [∀ (i : ι), order_topology (π i)] {a x : Π (i : ι), π i} [nonempty ι] (ha : ∀ (i : ι), x i < a i) :

set.Iio a ∈ nhds x

theorem pi_Iio_mem_nhds' {α : Type u} [topological_space α] [linear_order α] [order_topology α] {ι : Type u_1} [finite ι] {a' x' : ι → α} [nonempty ι] (ha : ∀ (i : ι), x' i < a' i) :

set.Iio a' ∈ nhds x'

theorem pi_Ioi_mem_nhds {ι : Type u_1} {π : ι → Type u_2} [finite ι] [Π (i : ι), linear_order (π i)] [Π (i : ι), topological_space (π i)] [∀ (i : ι), order_topology (π i)] {a x : Π (i : ι), π i} [nonempty ι] (ha : ∀ (i : ι), a i < x i) :

set.Ioi a ∈ nhds x

theorem pi_Ioi_mem_nhds' {α : Type u} [topological_space α] [linear_order α] [order_topology α] {ι : Type u_1} [finite ι] {a' x' : ι → α} [nonempty ι] (ha : ∀ (i : ι), a' i < x' i) :

set.Ioi a' ∈ nhds x'

theorem pi_Ioc_mem_nhds {ι : Type u_1} {π : ι → Type u_2} [finite ι] [Π (i : ι), linear_order (π i)] [Π (i : ι), topological_space (π i)] [∀ (i : ι), order_topology (π i)] {a b x : Π (i : ι), π i} [nonempty ι] (ha : ∀ (i : ι), a i < x i) (hb : ∀ (i : ι), x i < b i) :

set.Ioc a b ∈ nhds x

theorem pi_Ioc_mem_nhds' {α : Type u} [topological_space α] [linear_order α] [order_topology α] {ι : Type u_1} [finite ι] {a' b' x' : ι → α} [nonempty ι] (ha : ∀ (i : ι), a' i < x' i) (hb : ∀ (i : ι), x' i < b' i) :

set.Ioc a' b' ∈ nhds x'

theorem pi_Ico_mem_nhds {ι : Type u_1} {π : ι → Type u_2} [finite ι] [Π (i : ι), linear_order (π i)] [Π (i : ι), topological_space (π i)] [∀ (i : ι), order_topology (π i)] {a b x : Π (i : ι), π i} [nonempty ι] (ha : ∀ (i : ι), a i < x i) (hb : ∀ (i : ι), x i < b i) :

set.Ico a b ∈ nhds x

theorem pi_Ico_mem_nhds' {α : Type u} [topological_space α] [linear_order α] [order_topology α] {ι : Type u_1} [finite ι] {a' b' x' : ι → α} [nonempty ι] (ha : ∀ (i : ι), a' i < x' i) (hb : ∀ (i : ι), x' i < b' i) :

set.Ico a' b' ∈ nhds x'

theorem pi_Ioo_mem_nhds {ι : Type u_1} {π : ι → Type u_2} [finite ι] [Π (i : ι), linear_order (π i)] [Π (i : ι), topological_space (π i)] [∀ (i : ι), order_topology (π i)] {a b x : Π (i : ι), π i} [nonempty ι] (ha : ∀ (i : ι), a i < x i) (hb : ∀ (i : ι), x i < b i) :

set.Ioo a b ∈ nhds x

theorem pi_Ioo_mem_nhds' {α : Type u} [topological_space α] [linear_order α] [order_topology α] {ι : Type u_1} [finite ι] {a' b' x' : ι → α} [nonempty ι] (ha : ∀ (i : ι), a' i < x' i) (hb : ∀ (i : ι), x' i < b' i) :

set.Ioo a' b' ∈ nhds x'

theorem disjoint_nhds_at_top {α : Type u} [topological_space α] [linear_order α] [order_topology α] [no_max_order α] (x : α) :

disjoint (nhds x) filter.at_top

@[simp]

theorem inf_nhds_at_top {α : Type u} [topological_space α] [linear_order α] [order_topology α] [no_max_order α] (x : α) :

nhds x ⊓ filter.at_top = ⊥

theorem disjoint_nhds_at_bot {α : Type u} [topological_space α] [linear_order α] [order_topology α] [no_min_order α] (x : α) :

disjoint (nhds x) filter.at_bot

@[simp]

theorem inf_nhds_at_bot {α : Type u} [topological_space α] [linear_order α] [order_topology α] [no_min_order α] (x : α) :

nhds x ⊓ filter.at_bot = ⊥

theorem not_tendsto_nhds_of_tendsto_at_top {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_topology α] [no_max_order α] {F : filter β} [F.ne_bot] {f : β → α} (hf : filter.tendsto f F filter.at_top) (x : α) :

¬filter.tendsto f F (nhds x)

theorem not_tendsto_at_top_of_tendsto_nhds {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_topology α] [no_max_order α] {F : filter β} [F.ne_bot] {f : β → α} {x : α} (hf : filter.tendsto f F (nhds x)) :

¬filter.tendsto f F filter.at_top

theorem not_tendsto_nhds_of_tendsto_at_bot {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_topology α] [no_min_order α] {F : filter β} [F.ne_bot] {f : β → α} (hf : filter.tendsto f F filter.at_bot) (x : α) :

¬filter.tendsto f F (nhds x)

theorem not_tendsto_at_bot_of_tendsto_nhds {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_topology α] [no_min_order α] {F : filter β} [F.ne_bot] {f : β → α} {x : α} (hf : filter.tendsto f F (nhds x)) :

¬filter.tendsto f F filter.at_bot

Neighborhoods to the left and to the right on an `order_topology` #

We've seen some properties of left and right neighborhood of a point in an order_closed_topology. In an order_topology, such neighborhoods can be characterized as the sets containing suitable intervals to the right or to the left of a. We give now these characterizations.

theorem tfae_mem_nhds_within_Ioi {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a b : α} (hab : a < b) (s : set α) :

[s ∈ nhds_within a (set.Ioi a), s ∈ nhds_within a (set.Ioc a b), s ∈ nhds_within a (set.Ioo a b), ∃ (u : α) (H : u ∈ set.Ioc a b), set.Ioo a u ⊆ s, ∃ (u : α) (H : u ∈ set.Ioi a), set.Ioo a u ⊆ s].tfae

The following statements are equivalent:

s is a neighborhood of a within (a, +∞)
s is a neighborhood of a within (a, b]
s is a neighborhood of a within (a, b)
s includes (a, u) for some u ∈ (a, b]
s includes (a, u) for some u > a

theorem mem_nhds_within_Ioi_iff_exists_mem_Ioc_Ioo_subset {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a u' : α} {s : set α} (hu' : a < u') :

s ∈ nhds_within a (set.Ioi a) ↔ ∃ (u : α) (H : u ∈ set.Ioc a u'), set.Ioo a u ⊆ s

theorem mem_nhds_within_Ioi_iff_exists_Ioo_subset' {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a u' : α} {s : set α} (hu' : a < u') :

s ∈ nhds_within a (set.Ioi a) ↔ ∃ (u : α) (H : u ∈ set.Ioi a), set.Ioo a u ⊆ s

A set is a neighborhood of a within (a, +∞) if and only if it contains an interval (a, u) with a < u < u', provided a is not a top element.

theorem mem_nhds_within_Ioi_iff_exists_Ioo_subset {α : Type u} [topological_space α] [linear_order α] [order_topology α] [no_max_order α] {a : α} {s : set α} :

s ∈ nhds_within a (set.Ioi a) ↔ ∃ (u : α) (H : u ∈ set.Ioi a), set.Ioo a u ⊆ s

A set is a neighborhood of a within (a, +∞) if and only if it contains an interval (a, u) with a < u.

theorem mem_nhds_within_Ioi_iff_exists_Ioc_subset {α : Type u} [topological_space α] [linear_order α] [order_topology α] [no_max_order α] [densely_ordered α] {a : α} {s : set α} :

s ∈ nhds_within a (set.Ioi a) ↔ ∃ (u : α) (H : u ∈ set.Ioi a), set.Ioc a u ⊆ s

A set is a neighborhood of a within (a, +∞) if and only if it contains an interval (a, u] with a < u.

theorem tfae_mem_nhds_within_Iio {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a b : α} (h : a < b) (s : set α) :

[s ∈ nhds_within b (set.Iio b), s ∈ nhds_within b (set.Ico a b), s ∈ nhds_within b (set.Ioo a b), ∃ (l : α) (H : l ∈ set.Ico a b), set.Ioo l b ⊆ s, ∃ (l : α) (H : l ∈ set.Iio b), set.Ioo l b ⊆ s].tfae

The following statements are equivalent:

s is a neighborhood of b within (-∞, b)
s is a neighborhood of b within [a, b)
s is a neighborhood of b within (a, b)
s includes (l, b) for some l ∈ [a, b)
s includes (l, b) for some l < b

theorem mem_nhds_within_Iio_iff_exists_mem_Ico_Ioo_subset {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a l' : α} {s : set α} (hl' : l' < a) :

s ∈ nhds_within a (set.Iio a) ↔ ∃ (l : α) (H : l ∈ set.Ico l' a), set.Ioo l a ⊆ s

theorem mem_nhds_within_Iio_iff_exists_Ioo_subset' {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a l' : α} {s : set α} (hl' : l' < a) :

s ∈ nhds_within a (set.Iio a) ↔ ∃ (l : α) (H : l ∈ set.Iio a), set.Ioo l a ⊆ s

A set is a neighborhood of a within (-∞, a) if and only if it contains an interval (l, a) with l < a, provided a is not a bottom element.

theorem mem_nhds_within_Iio_iff_exists_Ioo_subset {α : Type u} [topological_space α] [linear_order α] [order_topology α] [no_min_order α] {a : α} {s : set α} :

s ∈ nhds_within a (set.Iio a) ↔ ∃ (l : α) (H : l ∈ set.Iio a), set.Ioo l a ⊆ s

A set is a neighborhood of a within (-∞, a) if and only if it contains an interval (l, a) with l < a.

theorem mem_nhds_within_Iio_iff_exists_Ico_subset {α : Type u} [topological_space α] [linear_order α] [order_topology α] [no_min_order α] [densely_ordered α] {a : α} {s : set α} :

s ∈ nhds_within a (set.Iio a) ↔ ∃ (l : α) (H : l ∈ set.Iio a), set.Ico l a ⊆ s

A set is a neighborhood of a within (-∞, a) if and only if it contains an interval [l, a) with l < a.

theorem tfae_mem_nhds_within_Ici {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a b : α} (hab : a < b) (s : set α) :

[s ∈ nhds_within a (set.Ici a), s ∈ nhds_within a (set.Icc a b), s ∈ nhds_within a (set.Ico a b), ∃ (u : α) (H : u ∈ set.Ioc a b), set.Ico a u ⊆ s, ∃ (u : α) (H : u ∈ set.Ioi a), set.Ico a u ⊆ s].tfae

The following statements are equivalent:

s is a neighborhood of a within [a, +∞)
s is a neighborhood of a within [a, b]
s is a neighborhood of a within [a, b)
s includes [a, u) for some u ∈ (a, b]
s includes [a, u) for some u > a

theorem mem_nhds_within_Ici_iff_exists_mem_Ioc_Ico_subset {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a u' : α} {s : set α} (hu' : a < u') :

s ∈ nhds_within a (set.Ici a) ↔ ∃ (u : α) (H : u ∈ set.Ioc a u'), set.Ico a u ⊆ s

theorem mem_nhds_within_Ici_iff_exists_Ico_subset' {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a u' : α} {s : set α} (hu' : a < u') :

s ∈ nhds_within a (set.Ici a) ↔ ∃ (u : α) (H : u ∈ set.Ioi a), set.Ico a u ⊆ s

A set is a neighborhood of a within [a, +∞) if and only if it contains an interval [a, u) with a < u < u', provided a is not a top element.

theorem mem_nhds_within_Ici_iff_exists_Ico_subset {α : Type u} [topological_space α] [linear_order α] [order_topology α] [no_max_order α] {a : α} {s : set α} :

s ∈ nhds_within a (set.Ici a) ↔ ∃ (u : α) (H : u ∈ set.Ioi a), set.Ico a u ⊆ s

A set is a neighborhood of a within [a, +∞) if and only if it contains an interval [a, u) with a < u.

theorem nhds_within_Ici_basis_Ico {α : Type u} [topological_space α] [linear_order α] [order_topology α] [no_max_order α] (a : α) :

(nhds_within a (set.Ici a)).has_basis (λ (u : α), a < u) (set.Ico a)

theorem mem_nhds_within_Ici_iff_exists_Icc_subset {α : Type u} [topological_space α] [linear_order α] [order_topology α] [no_max_order α] [densely_ordered α] {a : α} {s : set α} :

s ∈ nhds_within a (set.Ici a) ↔ ∃ (u : α), a < u ∧ set.Icc a u ⊆ s

A set is a neighborhood of a within [a, +∞) if and only if it contains an interval [a, u] with a < u.

theorem tfae_mem_nhds_within_Iic {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a b : α} (h : a < b) (s : set α) :

[s ∈ nhds_within b (set.Iic b), s ∈ nhds_within b (set.Icc a b), s ∈ nhds_within b (set.Ioc a b), ∃ (l : α) (H : l ∈ set.Ico a b), set.Ioc l b ⊆ s, ∃ (l : α) (H : l ∈ set.Iio b), set.Ioc l b ⊆ s].tfae

The following statements are equivalent:

s is a neighborhood of b within (-∞, b]
s is a neighborhood of b within [a, b]
s is a neighborhood of b within (a, b]
s includes (l, b] for some l ∈ [a, b)
s includes (l, b] for some l < b

theorem mem_nhds_within_Iic_iff_exists_mem_Ico_Ioc_subset {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a l' : α} {s : set α} (hl' : l' < a) :

s ∈ nhds_within a (set.Iic a) ↔ ∃ (l : α) (H : l ∈ set.Ico l' a), set.Ioc l a ⊆ s

theorem mem_nhds_within_Iic_iff_exists_Ioc_subset' {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a l' : α} {s : set α} (hl' : l' < a) :

s ∈ nhds_within a (set.Iic a) ↔ ∃ (l : α) (H : l ∈ set.Iio a), set.Ioc l a ⊆ s

A set is a neighborhood of a within (-∞, a] if and only if it contains an interval (l, a] with l < a, provided a is not a bottom element.

theorem mem_nhds_within_Iic_iff_exists_Ioc_subset {α : Type u} [topological_space α] [linear_order α] [order_topology α] [no_min_order α] {a : α} {s : set α} :

s ∈ nhds_within a (set.Iic a) ↔ ∃ (l : α) (H : l ∈ set.Iio a), set.Ioc l a ⊆ s

A set is a neighborhood of a within (-∞, a] if and only if it contains an interval (l, a] with l < a.

theorem mem_nhds_within_Iic_iff_exists_Icc_subset {α : Type u} [topological_space α] [linear_order α] [order_topology α] [no_min_order α] [densely_ordered α] {a : α} {s : set α} :

s ∈ nhds_within a (set.Iic a) ↔ ∃ (l : α), l < a ∧ set.Icc l a ⊆ s

A set is a neighborhood of a within (-∞, a] if and only if it contains an interval [l, a] with l < a.

theorem nhds_eq_infi_abs_sub {α : Type u} [topological_space α] [linear_ordered_add_comm_group α] [order_topology α] (a : α) :

nhds a = ⨅ (r : α) (H : r > 0), filter.principal {b : α | |a - b| < r}

theorem order_topology_of_nhds_abs {α : Type u_1} [topological_space α] [linear_ordered_add_comm_group α] (h_nhds : ∀ (a : α), nhds a = ⨅ (r : α) (H : r > 0), filter.principal {b : α | |a - b| < r}) :

order_topology α

theorem linear_ordered_add_comm_group.tendsto_nhds {α : Type u} {β : Type v} [topological_space α] [linear_ordered_add_comm_group α] [order_topology α] {f : β → α} {x : filter β} {a : α} :

filter.tendsto f x (nhds a) ↔ ∀ (ε : α), ε > 0 → (∀ᶠ (b : β) in x, |f b - a| < ε)

theorem eventually_abs_sub_lt {α : Type u} [topological_space α] [linear_ordered_add_comm_group α] [order_topology α] (a : α) {ε : α} (hε : 0 < ε) :

∀ᶠ (x : α) in nhds a, |x - a| < ε

theorem filter.tendsto.add_at_top {α : Type u} {β : Type v} [topological_space α] [linear_ordered_add_comm_group α] [order_topology α] {l : filter β} {f g : β → α} {C : α} (hf : filter.tendsto f l (nhds C)) (hg : filter.tendsto g l filter.at_top) :

filter.tendsto (λ (x : β), f x + g x) l filter.at_top

In a linearly ordered additive commutative group with the order topology, if f tends to C and g tends to at_top then f + g tends to at_top.

theorem filter.tendsto.add_at_bot {α : Type u} {β : Type v} [topological_space α] [linear_ordered_add_comm_group α] [order_topology α] {l : filter β} {f g : β → α} {C : α} (hf : filter.tendsto f l (nhds C)) (hg : filter.tendsto g l filter.at_bot) :

filter.tendsto (λ (x : β), f x + g x) l filter.at_bot

In a linearly ordered additive commutative group with the order topology, if f tends to C and g tends to at_bot then f + g tends to at_bot.

theorem filter.tendsto.at_top_add {α : Type u} {β : Type v} [topological_space α] [linear_ordered_add_comm_group α] [order_topology α] {l : filter β} {f g : β → α} {C : α} (hf : filter.tendsto f l filter.at_top) (hg : filter.tendsto g l (nhds C)) :

filter.tendsto (λ (x : β), f x + g x) l filter.at_top

In a linearly ordered additive commutative group with the order topology, if f tends to at_top and g tends to C then f + g tends to at_top.

theorem filter.tendsto.at_bot_add {α : Type u} {β : Type v} [topological_space α] [linear_ordered_add_comm_group α] [order_topology α] {l : filter β} {f g : β → α} {C : α} (hf : filter.tendsto f l filter.at_bot) (hg : filter.tendsto g l (nhds C)) :

filter.tendsto (λ (x : β), f x + g x) l filter.at_bot

In a linearly ordered additive commutative group with the order topology, if f tends to at_bot and g tends to C then f + g tends to at_bot.

theorem nhds_basis_Ioo_pos {α : Type u} [topological_space α] [linear_ordered_add_comm_group α] [order_topology α] [no_min_order α] [no_max_order α] (a : α) :

(nhds a).has_basis (λ (ε : α), 0 < ε) (λ (ε : α), set.Ioo (a - ε) (a + ε))

theorem nhds_basis_abs_sub_lt {α : Type u} [topological_space α] [linear_ordered_add_comm_group α] [order_topology α] [no_min_order α] [no_max_order α] (a : α) :

(nhds a).has_basis (λ (ε : α), 0 < ε) (λ (ε : α), {b : α | |b - a| < ε})

theorem nhds_basis_zero_abs_sub_lt (α : Type u) [topological_space α] [linear_ordered_add_comm_group α] [order_topology α] [no_min_order α] [no_max_order α] :

(nhds 0).has_basis (λ (ε : α), 0 < ε) (λ (ε : α), {b : α | |b| < ε})

theorem nhds_basis_Ioo_pos_of_pos {α : Type u} [topological_space α] [linear_ordered_add_comm_group α] [order_topology α] [no_min_order α] [no_max_order α] {a : α} (ha : 0 < a) :

(nhds a).has_basis (λ (ε : α), 0 < ε ∧ ε ≤ a) (λ (ε : α), set.Ioo (a - ε) (a + ε))

If a is positive we can form a basis from only nonnegative Ioo intervals

theorem preimage_neg {α : Type u} [add_group α] :

set.preimage has_neg.neg = set.image has_neg.neg

theorem filter.map_neg_eq_comap_neg {α : Type u} [add_group α] :

filter.map has_neg.neg = filter.comap has_neg.neg

theorem is_lub.frequently_mem {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a : α} {s : set α} (ha : is_lub s a) (hs : s.nonempty) :

∃ᶠ (x : α) in nhds_within a (set.Iic a), x ∈ s

theorem is_lub.frequently_nhds_mem {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a : α} {s : set α} (ha : is_lub s a) (hs : s.nonempty) :

∃ᶠ (x : α) in nhds a, x ∈ s

theorem is_glb.frequently_mem {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a : α} {s : set α} (ha : is_glb s a) (hs : s.nonempty) :

∃ᶠ (x : α) in nhds_within a (set.Ici a), x ∈ s

theorem is_glb.frequently_nhds_mem {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a : α} {s : set α} (ha : is_glb s a) (hs : s.nonempty) :

∃ᶠ (x : α) in nhds a, x ∈ s

theorem is_lub.mem_closure {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a : α} {s : set α} (ha : is_lub s a) (hs : s.nonempty) :

a ∈ closure s

theorem is_glb.mem_closure {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a : α} {s : set α} (ha : is_glb s a) (hs : s.nonempty) :

a ∈ closure s

theorem is_lub.nhds_within_ne_bot {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a : α} {s : set α} (ha : is_lub s a) (hs : s.nonempty) :

(nhds_within a s).ne_bot

theorem is_glb.nhds_within_ne_bot {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a : α} {s : set α} :

is_glb s a → s.nonempty → (nhds_within a s).ne_bot

theorem is_lub_of_mem_nhds {α : Type u} [topological_space α] [linear_order α] [order_topology α] {s : set α} {a : α} {f : filter α} (hsa : a ∈ upper_bounds s) (hsf : s ∈ f) [(f ⊓ nhds a).ne_bot] :

theorem is_lub_of_mem_closure {α : Type u} [topological_space α] [linear_order α] [order_topology α] {s : set α} {a : α} (hsa : a ∈ upper_bounds s) (hsf : a ∈ closure s) :

theorem is_glb_of_mem_nhds {α : Type u} [topological_space α] [linear_order α] [order_topology α] {s : set α} {a : α} {f : filter α} :

a ∈ lower_bounds s → s ∈ f → (f ⊓ nhds a).ne_bot → is_glb s a

theorem is_glb_of_mem_closure {α : Type u} [topological_space α] [linear_order α] [order_topology α] {s : set α} {a : α} (hsa : a ∈ lower_bounds s) (hsf : a ∈ closure s) :

theorem is_lub.mem_upper_bounds_of_tendsto {α : Type u} {γ : Type w} [topological_space α] [linear_order α] [order_topology α] [preorder γ] [topological_space γ] [order_closed_topology γ] {f : α → γ} {s : set α} {a : α} {b : γ} (hf : monotone_on f s) (ha : is_lub s a) (hb : filter.tendsto f (nhds_within a s) (nhds b)) :

b ∈ upper_bounds (f '' s)

theorem is_lub.is_lub_of_tendsto {α : Type u} {γ : Type w} [topological_space α] [linear_order α] [order_topology α] [preorder γ] [topological_space γ] [order_closed_topology γ] {f : α → γ} {s : set α} {a : α} {b : γ} (hf : monotone_on f s) (ha : is_lub s a) (hs : s.nonempty) (hb : filter.tendsto f (nhds_within a s) (nhds b)) :

is_lub (f '' s) b

theorem is_glb.mem_lower_bounds_of_tendsto {α : Type u} {γ : Type w} [topological_space α] [linear_order α] [order_topology α] [preorder γ] [topological_space γ] [order_closed_topology γ] {f : α → γ} {s : set α} {a : α} {b : γ} (hf : monotone_on f s) (ha : is_glb s a) (hb : filter.tendsto f (nhds_within a s) (nhds b)) :

b ∈ lower_bounds (f '' s)

theorem is_glb.is_glb_of_tendsto {α : Type u} {γ : Type w} [topological_space α] [linear_order α] [order_topology α] [preorder γ] [topological_space γ] [order_closed_topology γ] {f : α → γ} {s : set α} {a : α} {b : γ} (hf : monotone_on f s) :

is_glb s a → s.nonempty → filter.tendsto f (nhds_within a s) (nhds b) → is_glb (f '' s) b

theorem is_lub.mem_lower_bounds_of_tendsto {α : Type u} {γ : Type w} [topological_space α] [linear_order α] [order_topology α] [preorder γ] [topological_space γ] [order_closed_topology γ] {f : α → γ} {s : set α} {a : α} {b : γ} (hf : antitone_on f s) (ha : is_lub s a) (hb : filter.tendsto f (nhds_within a s) (nhds b)) :

b ∈ lower_bounds (f '' s)

theorem is_lub.is_glb_of_tendsto {α : Type u} {γ : Type w} [topological_space α] [linear_order α] [order_topology α] [preorder γ] [topological_space γ] [order_closed_topology γ] {f : α → γ} {s : set α} {a : α} {b : γ} :

antitone_on f s → is_lub s a → s.nonempty → filter.tendsto f (nhds_within a s) (nhds b) → is_glb (f '' s) b

theorem is_glb.mem_upper_bounds_of_tendsto {α : Type u} {γ : Type w} [topological_space α] [linear_order α] [order_topology α] [preorder γ] [topological_space γ] [order_closed_topology γ] {f : α → γ} {s : set α} {a : α} {b : γ} (hf : antitone_on f s) (ha : is_glb s a) (hb : filter.tendsto f (nhds_within a s) (nhds b)) :

b ∈ upper_bounds (f '' s)

theorem is_glb.is_lub_of_tendsto {α : Type u} {γ : Type w} [topological_space α] [linear_order α] [order_topology α] [preorder γ] [topological_space γ] [order_closed_topology γ] {f : α → γ} {s : set α} {a : α} {b : γ} :

antitone_on f s → is_glb s a → s.nonempty → filter.tendsto f (nhds_within a s) (nhds b) → is_lub (f '' s) b

theorem is_lub.mem_of_is_closed {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a : α} {s : set α} (ha : is_lub s a) (hs : s.nonempty) (sc : is_closed s) :

a ∈ s

theorem is_closed.is_lub_mem {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a : α} {s : set α} (ha : is_lub s a) (hs : s.nonempty) (sc : is_closed s) :

a ∈ s

Alias of is_lub.mem_of_is_closed.

theorem is_glb.mem_of_is_closed {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a : α} {s : set α} (ha : is_glb s a) (hs : s.nonempty) (sc : is_closed s) :

a ∈ s

theorem is_closed.is_glb_mem {α : Type u} [topological_space α] [linear_order α] [order_topology α] {a : α} {s : set α} (ha : is_glb s a) (hs : s.nonempty) (sc : is_closed s) :

a ∈ s

Alias of is_glb.mem_of_is_closed.

Existence of sequences tending to Inf or Sup of a given set #

theorem is_lub.exists_seq_strict_mono_tendsto_of_not_mem {α : Type u} [topological_space α] [linear_order α] [order_topology α] {t : set α} {x : α} [(nhds x).is_countably_generated] (htx : is_lub t x) (not_mem : x ∉ t) (ht : t.nonempty) :

∃ (u : ℕ → α), strict_mono u ∧ (∀ (n : ℕ), u n < x) ∧ filter.tendsto u filter.at_top (nhds x) ∧ ∀ (n : ℕ), u n ∈ t

theorem is_lub.exists_seq_monotone_tendsto {α : Type u} [topological_space α] [linear_order α] [order_topology α] {t : set α} {x : α} [(nhds x).is_countably_generated] (htx : is_lub t x) (ht : t.nonempty) :

∃ (u : ℕ → α), monotone u ∧ (∀ (n : ℕ), u n ≤ x) ∧ filter.tendsto u filter.at_top (nhds x) ∧ ∀ (n : ℕ), u n ∈ t

theorem exists_seq_strict_mono_tendsto' {α : Type u_1} [linear_order α] [topological_space α] [densely_ordered α] [order_topology α] [topological_space.first_countable_topology α] {x y : α} (hy : y < x) :

∃ (u : ℕ → α), strict_mono u ∧ (∀ (n : ℕ), u n ∈ set.Ioo y x) ∧ filter.tendsto u filter.at_top (nhds x)

theorem exists_seq_strict_mono_tendsto {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] [no_min_order α] [topological_space.first_countable_topology α] (x : α) :

∃ (u : ℕ → α), strict_mono u ∧ (∀ (n : ℕ), u n < x) ∧ filter.tendsto u filter.at_top (nhds x)

theorem exists_seq_strict_mono_tendsto_nhds_within {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] [no_min_order α] [topological_space.first_countable_topology α] (x : α) :

∃ (u : ℕ → α), strict_mono u ∧ (∀ (n : ℕ), u n < x) ∧ filter.tendsto u filter.at_top (nhds_within x (set.Iio x))

theorem exists_seq_tendsto_Sup {α : Type u_1} [conditionally_complete_linear_order α] [topological_space α] [order_topology α] [topological_space.first_countable_topology α] {S : set α} (hS : S.nonempty) (hS' : bdd_above S) :

∃ (u : ℕ → α), monotone u ∧ filter.tendsto u filter.at_top (nhds (has_Sup.Sup S)) ∧ ∀ (n : ℕ), u n ∈ S

theorem is_glb.exists_seq_strict_anti_tendsto_of_not_mem {α : Type u} [topological_space α] [linear_order α] [order_topology α] {t : set α} {x : α} [(nhds x).is_countably_generated] (htx : is_glb t x) (not_mem : x ∉ t) (ht : t.nonempty) :

∃ (u : ℕ → α), strict_anti u ∧ (∀ (n : ℕ), x < u n) ∧ filter.tendsto u filter.at_top (nhds x) ∧ ∀ (n : ℕ), u n ∈ t

theorem is_glb.exists_seq_antitone_tendsto {α : Type u} [topological_space α] [linear_order α] [order_topology α] {t : set α} {x : α} [(nhds x).is_countably_generated] (htx : is_glb t x) (ht : t.nonempty) :

∃ (u : ℕ → α), antitone u ∧ (∀ (n : ℕ), x ≤ u n) ∧ filter.tendsto u filter.at_top (nhds x) ∧ ∀ (n : ℕ), u n ∈ t

theorem exists_seq_strict_anti_tendsto' {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] [topological_space.first_countable_topology α] {x y : α} (hy : x < y) :

∃ (u : ℕ → α), strict_anti u ∧ (∀ (n : ℕ), u n ∈ set.Ioo x y) ∧ filter.tendsto u filter.at_top (nhds x)

theorem exists_seq_strict_anti_tendsto {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] [no_max_order α] [topological_space.first_countable_topology α] (x : α) :

∃ (u : ℕ → α), strict_anti u ∧ (∀ (n : ℕ), x < u n) ∧ filter.tendsto u filter.at_top (nhds x)

theorem exists_seq_strict_anti_tendsto_nhds_within {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] [no_max_order α] [topological_space.first_countable_topology α] (x : α) :

∃ (u : ℕ → α), strict_anti u ∧ (∀ (n : ℕ), x < u n) ∧ filter.tendsto u filter.at_top (nhds_within x (set.Ioi x))

theorem exists_seq_strict_anti_strict_mono_tendsto {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] [topological_space.first_countable_topology α] {x y : α} (h : x < y) :

∃ (u v : ℕ → α), strict_anti u ∧ strict_mono v ∧ (∀ (k : ℕ), u k ∈ set.Ioo x y) ∧ (∀ (l : ℕ), v l ∈ set.Ioo x y) ∧ (∀ (k l : ℕ), u k < v l) ∧ filter.tendsto u filter.at_top (nhds x) ∧ filter.tendsto v filter.at_top (nhds y)

theorem exists_seq_tendsto_Inf {α : Type u_1} [conditionally_complete_linear_order α] [topological_space α] [order_topology α] [topological_space.first_countable_topology α] {S : set α} (hS : S.nonempty) (hS' : bdd_below S) :

∃ (u : ℕ → α), antitone u ∧ filter.tendsto u filter.at_top (nhds (has_Inf.Inf S)) ∧ ∀ (n : ℕ), u n ∈ S

theorem closure_Ioi' {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {a : α} (h : (set.Ioi a).nonempty) :

closure (set.Ioi a) = set.Ici a

The closure of the interval (a, +∞) is the closed interval [a, +∞), unless a is a top element.

@[simp]

theorem closure_Ioi {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] (a : α) [no_max_order α] :

closure (set.Ioi a) = set.Ici a

The closure of the interval (a, +∞) is the closed interval [a, +∞).

theorem closure_Iio' {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {a : α} (h : (set.Iio a).nonempty) :

closure (set.Iio a) = set.Iic a

The closure of the interval (-∞, a) is the closed interval (-∞, a], unless a is a bottom element.

@[simp]

theorem closure_Iio {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] (a : α) [no_min_order α] :

closure (set.Iio a) = set.Iic a

The closure of the interval (-∞, a) is the interval (-∞, a].

@[simp]

theorem closure_Ioo {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {a b : α} (hab : a ≠ b) :

closure (set.Ioo a b) = set.Icc a b

The closure of the open interval (a, b) is the closed interval [a, b].

@[simp]

theorem closure_Ioc {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {a b : α} (hab : a ≠ b) :

closure (set.Ioc a b) = set.Icc a b

The closure of the interval (a, b] is the closed interval [a, b].

@[simp]

theorem closure_Ico {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {a b : α} (hab : a ≠ b) :

closure (set.Ico a b) = set.Icc a b

The closure of the interval [a, b) is the closed interval [a, b].

@[simp]

theorem interior_Ici' {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {a : α} (ha : (set.Iio a).nonempty) :

interior (set.Ici a) = set.Ioi a

theorem interior_Ici {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] [no_min_order α] {a : α} :

interior (set.Ici a) = set.Ioi a

@[simp]

theorem interior_Iic' {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {a : α} (ha : (set.Ioi a).nonempty) :

interior (set.Iic a) = set.Iio a

theorem interior_Iic {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] [no_max_order α] {a : α} :

interior (set.Iic a) = set.Iio a

@[simp]

theorem interior_Icc {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] [no_min_order α] [no_max_order α] {a b : α} :

interior (set.Icc a b) = set.Ioo a b

@[simp]

theorem interior_Ico {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] [no_min_order α] {a b : α} :

interior (set.Ico a b) = set.Ioo a b

@[simp]

theorem interior_Ioc {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] [no_max_order α] {a b : α} :

interior (set.Ioc a b) = set.Ioo a b

theorem closure_interior_Icc {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {a b : α} (h : a ≠ b) :

closure (interior (set.Icc a b)) = set.Icc a b

theorem Ioc_subset_closure_interior {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] (a b : α) :

set.Ioc a b ⊆ closure (interior (set.Ioc a b))

theorem Ico_subset_closure_interior {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] (a b : α) :

set.Ico a b ⊆ closure (interior (set.Ico a b))

@[simp]

theorem frontier_Ici' {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {a : α} (ha : (set.Iio a).nonempty) :

frontier (set.Ici a) = {a}

theorem frontier_Ici {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] [no_min_order α] {a : α} :

frontier (set.Ici a) = {a}

@[simp]

theorem frontier_Iic' {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {a : α} (ha : (set.Ioi a).nonempty) :

frontier (set.Iic a) = {a}

theorem frontier_Iic {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] [no_max_order α] {a : α} :

frontier (set.Iic a) = {a}

@[simp]

theorem frontier_Ioi' {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {a : α} (ha : (set.Ioi a).nonempty) :

frontier (set.Ioi a) = {a}

theorem frontier_Ioi {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] [no_max_order α] {a : α} :

frontier (set.Ioi a) = {a}

@[simp]

theorem frontier_Iio' {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {a : α} (ha : (set.Iio a).nonempty) :

frontier (set.Iio a) = {a}

theorem frontier_Iio {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] [no_min_order α] {a : α} :

frontier (set.Iio a) = {a}

@[simp]

theorem frontier_Icc {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] [no_min_order α] [no_max_order α] {a b : α} (h : a ≤ b) :

frontier (set.Icc a b) = {a, b}

@[simp]

theorem frontier_Ioo {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {a b : α} (h : a < b) :

frontier (set.Ioo a b) = {a, b}

@[simp]

theorem frontier_Ico {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] [no_min_order α] {a b : α} (h : a < b) :

frontier (set.Ico a b) = {a, b}

@[simp]

theorem frontier_Ioc {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] [no_max_order α] {a b : α} (h : a < b) :

frontier (set.Ioc a b) = {a, b}

theorem nhds_within_Ioi_ne_bot' {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {a b : α} (H₁ : (set.Ioi a).nonempty) (H₂ : a ≤ b) :

(nhds_within b (set.Ioi a)).ne_bot

theorem nhds_within_Ioi_ne_bot {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] [no_max_order α] {a b : α} (H : a ≤ b) :

(nhds_within b (set.Ioi a)).ne_bot

theorem nhds_within_Ioi_self_ne_bot' {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {a : α} (H : (set.Ioi a).nonempty) :

(nhds_within a (set.Ioi a)).ne_bot

@[instance]

theorem nhds_within_Ioi_self_ne_bot {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] [no_max_order α] (a : α) :

(nhds_within a (set.Ioi a)).ne_bot

theorem filter.eventually.exists_gt {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] [no_max_order α] {a : α} {p : α → Prop} (h : ∀ᶠ (x : α) in nhds a, p x) :

∃ (b : α) (H : b > a), p b

theorem nhds_within_Iio_ne_bot' {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {b c : α} (H₁ : (set.Iio c).nonempty) (H₂ : b ≤ c) :

(nhds_within b (set.Iio c)).ne_bot

theorem nhds_within_Iio_ne_bot {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] [no_min_order α] {a b : α} (H : a ≤ b) :

(nhds_within a (set.Iio b)).ne_bot

theorem nhds_within_Iio_self_ne_bot' {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {b : α} (H : (set.Iio b).nonempty) :

(nhds_within b (set.Iio b)).ne_bot

@[instance]

theorem nhds_within_Iio_self_ne_bot {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] [no_min_order α] (a : α) :

(nhds_within a (set.Iio a)).ne_bot

theorem filter.eventually.exists_lt {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] [no_min_order α] {a : α} {p : α → Prop} (h : ∀ᶠ (x : α) in nhds a, p x) :

∃ (b : α) (H : b < a), p b

theorem right_nhds_within_Ico_ne_bot {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {a b : α} (H : a < b) :

(nhds_within b (set.Ico a b)).ne_bot

theorem left_nhds_within_Ioc_ne_bot {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {a b : α} (H : a < b) :

(nhds_within a (set.Ioc a b)).ne_bot

theorem left_nhds_within_Ioo_ne_bot {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {a b : α} (H : a < b) :

(nhds_within a (set.Ioo a b)).ne_bot

theorem right_nhds_within_Ioo_ne_bot {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {a b : α} (H : a < b) :

(nhds_within b (set.Ioo a b)).ne_bot

theorem comap_coe_nhds_within_Iio_of_Ioo_subset {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {b : α} {s : set α} (hb : s ⊆ set.Iio b) (hs : s.nonempty → (∃ (a : α) (H : a < b), set.Ioo a b ⊆ s)) :

filter.comap coe (nhds_within b (set.Iio b)) = filter.at_top

theorem comap_coe_nhds_within_Ioi_of_Ioo_subset {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {a : α} {s : set α} (ha : s ⊆ set.Ioi a) (hs : s.nonempty → (∃ (b : α) (H : b > a), set.Ioo a b ⊆ s)) :

filter.comap coe (nhds_within a (set.Ioi a)) = filter.at_bot

theorem map_coe_at_top_of_Ioo_subset {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {b : α} {s : set α} (hb : s ⊆ set.Iio b) (hs : ∀ (a' : α), a' < b → (∃ (a : α) (H : a < b), set.Ioo a b ⊆ s)) :

filter.map coe filter.at_top = nhds_within b (set.Iio b)

theorem map_coe_at_bot_of_Ioo_subset {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {a : α} {s : set α} (ha : s ⊆ set.Ioi a) (hs : ∀ (b' : α), b' > a → (∃ (b : α) (H : b > a), set.Ioo a b ⊆ s)) :

filter.map coe filter.at_bot = nhds_within a (set.Ioi a)

theorem comap_coe_Ioo_nhds_within_Iio {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] (a b : α) :

filter.comap coe (nhds_within b (set.Iio b)) = filter.at_top

The at_top filter for an open interval Ioo a b comes from the left-neighbourhoods filter at the right endpoint in the ambient order.

theorem comap_coe_Ioo_nhds_within_Ioi {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] (a b : α) :

filter.comap coe (nhds_within a (set.Ioi a)) = filter.at_bot

The at_bot filter for an open interval Ioo a b comes from the right-neighbourhoods filter at the left endpoint in the ambient order.

theorem comap_coe_Ioi_nhds_within_Ioi {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] (a : α) :

filter.comap coe (nhds_within a (set.Ioi a)) = filter.at_bot

theorem comap_coe_Iio_nhds_within_Iio {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] (a : α) :

filter.comap coe (nhds_within a (set.Iio a)) = filter.at_top

@[simp]

theorem map_coe_Ioo_at_top {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {a b : α} (h : a < b) :

filter.map coe filter.at_top = nhds_within b (set.Iio b)

@[simp]

theorem map_coe_Ioo_at_bot {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {a b : α} (h : a < b) :

filter.map coe filter.at_bot = nhds_within a (set.Ioi a)

@[simp]

theorem map_coe_Ioi_at_bot {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] (a : α) :

filter.map coe filter.at_bot = nhds_within a (set.Ioi a)

@[simp]

theorem map_coe_Iio_at_top {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] (a : α) :

filter.map coe filter.at_top = nhds_within a (set.Iio a)

@[simp]

theorem tendsto_comp_coe_Ioo_at_top {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {a b : α} {l : filter β} {f : α → β} (h : a < b) :

filter.tendsto (λ (x : ↥(set.Ioo a b)), f ↑x) filter.at_top l ↔ filter.tendsto f (nhds_within b (set.Iio b)) l

@[simp]

theorem tendsto_comp_coe_Ioo_at_bot {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {a b : α} {l : filter β} {f : α → β} (h : a < b) :

filter.tendsto (λ (x : ↥(set.Ioo a b)), f ↑x) filter.at_bot l ↔ filter.tendsto f (nhds_within a (set.Ioi a)) l

@[simp]

theorem tendsto_comp_coe_Ioi_at_bot {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {a : α} {l : filter β} {f : α → β} :

filter.tendsto (λ (x : ↥(set.Ioi a)), f ↑x) filter.at_bot l ↔ filter.tendsto f (nhds_within a (set.Ioi a)) l

@[simp]

theorem tendsto_comp_coe_Iio_at_top {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {a : α} {l : filter β} {f : α → β} :

filter.tendsto (λ (x : ↥(set.Iio a)), f ↑x) filter.at_top l ↔ filter.tendsto f (nhds_within a (set.Iio a)) l

@[simp]

theorem tendsto_Ioo_at_top {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {a b : α} {l : filter β} {f : β → ↥(set.Ioo a b)} :

filter.tendsto f l filter.at_top ↔ filter.tendsto (λ (x : β), ↑(f x)) l (nhds_within b (set.Iio b))

@[simp]

theorem tendsto_Ioo_at_bot {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {a b : α} {l : filter β} {f : β → ↥(set.Ioo a b)} :

filter.tendsto f l filter.at_bot ↔ filter.tendsto (λ (x : β), ↑(f x)) l (nhds_within a (set.Ioi a))

@[simp]

theorem tendsto_Ioi_at_bot {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {a : α} {l : filter β} {f : β → ↥(set.Ioi a)} :

filter.tendsto f l filter.at_bot ↔ filter.tendsto (λ (x : β), ↑(f x)) l (nhds_within a (set.Ioi a))

@[simp]

theorem tendsto_Iio_at_top {α : Type u} {β : Type v} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] {a : α} {l : filter β} {f : β → ↥(set.Iio a)} :

filter.tendsto f l filter.at_top ↔ filter.tendsto (λ (x : β), ↑(f x)) l (nhds_within a (set.Iio a))

@[protected, instance]

def nhds_within.filter.ne_bot {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] (x : α) [nontrivial α] :

(nhds_within x {x}ᶜ).ne_bot

theorem dense.exists_countable_dense_subset_no_bot_top {α : Type u} [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] [nontrivial α] {s : set α} [topological_space.separable_space ↥s] (hs : dense s) :

∃ (t : set α) (H : t ⊆ s), t.countable ∧ dense t ∧ (∀ (x : α), is_bot x → x ∉ t) ∧ ∀ (x : α), is_top x → x ∉ t

Let s be a dense set in a nontrivial dense linear order α. If s is a separable space (e.g., if α has a second countable topology), then there exists a countable dense subset t ⊆ s such that t does not contain bottom/top elements of α.

theorem exists_countable_dense_no_bot_top (α : Type u) [topological_space α] [linear_order α] [order_topology α] [densely_ordered α] [topological_space.separable_space α] [nontrivial α] :

∃ (s : set α), s.countable ∧ dense s ∧ (∀ (x : α), is_bot x → x ∉ s) ∧ ∀ (x : α), is_top x → x ∉ s

If α is a nontrivial separable dense linear order, then there exists a countable dense set s : set α that contains neither top nor bottom elements of α. For a dense set containing both bot and top elements, see exists_countable_dense_bot_top.

theorem Sup_mem_closure {α : Type u} [topological_space α] [complete_linear_order α] [order_topology α] {s : set α} (hs : s.nonempty) :

has_Sup.Sup s ∈ closure s

theorem Inf_mem_closure {α : Type u} [topological_space α] [complete_linear_order α] [order_topology α] {s : set α} (hs : s.nonempty) :

has_Inf.Inf s ∈ closure s

theorem is_closed.Sup_mem {α : Type u} [topological_space α] [complete_linear_order α] [order_topology α] {s : set α} (hs : s.nonempty) (hc : is_closed s) :

has_Sup.Sup s ∈ s

theorem is_closed.Inf_mem {α : Type u} [topological_space α] [complete_linear_order α] [order_topology α] {s : set α} (hs : s.nonempty) (hc : is_closed s) :

has_Inf.Inf s ∈ s

theorem monotone.map_Sup_of_continuous_at' {α : Type u} {β : Type v} [complete_linear_order α] [topological_space α] [order_topology α] [complete_linear_order β] [topological_space β] [order_closed_topology β] {f : α → β} {s : set α} (Cf : continuous_at f (has_Sup.Sup s)) (Mf : monotone f) (hs : s.nonempty) :

f (has_Sup.Sup s) = has_Sup.Sup (f '' s)

A monotone function continuous at the supremum of a nonempty set sends this supremum to the supremum of the image of this set.

theorem monotone.map_Sup_of_continuous_at {α : Type u} {β : Type v} [complete_linear_order α] [topological_space α] [order_topology α] [complete_linear_order β] [topological_space β] [order_closed_topology β] {f : α → β} {s : set α} (Cf : continuous_at f (has_Sup.Sup s)) (Mf : monotone f) (fbot : f ⊥ = ⊥) :

f (has_Sup.Sup s) = has_Sup.Sup (f '' s)

A monotone function f sending bot to bot and continuous at the supremum of a set sends this supremum to the supremum of the image of this set.

theorem monotone.map_supr_of_continuous_at' {α : Type u} {β : Type v} [complete_linear_order α] [topological_space α] [order_topology α] [complete_linear_order β] [topological_space β] [order_closed_topology β] {ι : Sort u_1} [nonempty ι] {f : α → β} {g : ι → α} (Cf : continuous_at f (supr g)) (Mf : monotone f) :

f (⨆ (i : ι), g i) = ⨆ (i : ι), f (g i)

A monotone function continuous at the indexed supremum over a nonempty Sort sends this indexed supremum to the indexed supremum of the composition.

theorem monotone.map_supr_of_continuous_at {α : Type u} {β : Type v} [complete_linear_order α] [topological_space α] [order_topology α] [complete_linear_order β] [topological_space β] [order_closed_topology β] {ι : Sort u_1} {f : α → β} {g : ι → α} (Cf : continuous_at f (supr g)) (Mf : monotone f) (fbot : f ⊥ = ⊥) :

f (⨆ (i : ι), g i) = ⨆ (i : ι), f (g i)

If a monotone function sending bot to bot is continuous at the indexed supremum over a Sort, then it sends this indexed supremum to the indexed supremum of the composition.

theorem monotone.map_Inf_of_continuous_at' {α : Type u} {β : Type v} [complete_linear_order α] [topological_space α] [order_topology α] [complete_linear_order β] [topological_space β] [order_closed_topology β] {f : α → β} {s : set α} (Cf : continuous_at f (has_Inf.Inf s)) (Mf : monotone f) (hs : s.nonempty) :

f (has_Inf.Inf s) = has_Inf.Inf (f '' s)

A monotone function continuous at the infimum of a nonempty set sends this infimum to the infimum of the image of this set.

theorem monotone.map_Inf_of_continuous_at {α : Type u} {β : Type v} [complete_linear_order α] [topological_space α] [order_topology α] [complete_linear_order β] [topological_space β] [order_closed_topology β] {f : α → β} {s : set α} (Cf : continuous_at f (has_Inf.Inf s)) (Mf : monotone f) (ftop : f ⊤ = ⊤) :

f (has_Inf.Inf s) = has_Inf.Inf (f '' s)

A monotone function f sending top to top and continuous at the infimum of a set sends this infimum to the infimum of the image of this set.

theorem monotone.map_infi_of_continuous_at' {α : Type u} {β : Type v} [complete_linear_order α] [topological_space α] [order_topology α] [complete_linear_order β] [topological_space β] [order_closed_topology β] {ι : Sort u_1} [nonempty ι] {f : α → β} {g : ι → α} (Cf : continuous_at f (infi g)) (Mf : monotone f) :

f (⨅ (i : ι), g i) = ⨅ (i : ι), f (g i)

A monotone function continuous at the indexed infimum over a nonempty Sort sends this indexed infimum to the indexed infimum of the composition.

theorem monotone.map_infi_of_continuous_at {α : Type u} {β : Type v} [complete_linear_order α] [topological_space α] [order_topology α] [complete_linear_order β] [topological_space β] [order_closed_topology β] {ι : Sort u_1} {f : α → β} {g : ι → α} (Cf : continuous_at f (infi g)) (Mf : monotone f) (ftop : f ⊤ = ⊤) :

f (infi g) = infi (f ∘ g)

If a monotone function sending top to top is continuous at the indexed infimum over a Sort, then it sends this indexed infimum to the indexed infimum of the composition.

theorem antitone.map_Sup_of_continuous_at' {α : Type u} {β : Type v} [complete_linear_order α] [topological_space α] [order_topology α] [complete_linear_order β] [topological_space β] [order_closed_topology β] {f : α → β} {s : set α} (Cf : continuous_at f (has_Sup.Sup s)) (Af : antitone f) (hs : s.nonempty) :

f (has_Sup.Sup s) = has_Inf.Inf (f '' s)

An antitone function continuous at the supremum of a nonempty set sends this supremum to the infimum of the image of this set.

theorem antitone.map_Sup_of_continuous_at {α : Type u} {β : Type v} [complete_linear_order α] [topological_space α] [order_topology α] [complete_linear_order β] [topological_space β] [order_closed_topology β] {f : α → β} {s : set α} (Cf : continuous_at f (has_Sup.Sup s)) (Af : antitone f) (fbot : f ⊥ = ⊤) :

f (has_Sup.Sup s) = has_Inf.Inf (f '' s)

An antitone function f sending bot to top and continuous at the supremum of a set sends this supremum to the infimum of the image of this set.

theorem antitone.map_supr_of_continuous_at' {α : Type u} {β : Type v} [complete_linear_order α] [topological_space α] [order_topology α] [complete_linear_order β] [topological_space β] [order_closed_topology β] {ι : Sort u_1} [nonempty ι] {f : α → β} {g : ι → α} (Cf : continuous_at f (supr g)) (Af : antitone f) :

f (⨆ (i : ι), g i) = ⨅ (i : ι), f (g i)

An antitone function continuous at the indexed supremum over a nonempty Sort sends this indexed supremum to the indexed infimum of the composition.

theorem antitone.map_supr_of_continuous_at {α : Type u} {β : Type v} [complete_linear_order α] [topological_space α] [order_topology α] [complete_linear_order β] [topological_space β] [order_closed_topology β] {ι : Sort u_1} {f : α → β} {g : ι → α} (Cf : continuous_at f (supr g)) (Af : antitone f) (fbot : f ⊥ = ⊤) :

f (⨆ (i : ι), g i) = ⨅ (i : ι), f (g i)

An antitone function sending bot to top is continuous at the indexed supremum over a Sort, then it sends this indexed supremum to the indexed supremum of the composition.

theorem antitone.map_Inf_of_continuous_at' {α : Type u} {β : Type v} [complete_linear_order α] [topological_space α] [order_topology α] [complete_linear_order β] [topological_space β] [order_closed_topology β] {f : α → β} {s : set α} (Cf : continuous_at f (has_Inf.Inf s)) (Af : antitone f) (hs : s.nonempty) :

f (has_Inf.Inf s) = has_Sup.Sup (f '' s)

An antitone function continuous at the infimum of a nonempty set sends this infimum to the supremum of the image of this set.

theorem antitone.map_Inf_of_continuous_at {α : Type u} {β : Type v} [complete_linear_order α] [topological_space α] [order_topology α] [complete_linear_order β] [topological_space β] [order_closed_topology β] {f : α → β} {s : set α} (Cf : continuous_at f (has_Inf.Inf s)) (Af : antitone f) (ftop : f ⊤ = ⊥) :

f (has_Inf.Inf s) = has_Sup.Sup (f '' s)

An antitone function f sending top to bot and continuous at the infimum of a set sends this infimum to the supremum of the image of this set.

theorem antitone.map_infi_of_continuous_at' {α : Type u} {β : Type v} [complete_linear_order α] [topological_space α] [order_topology α] [complete_linear_order β] [topological_space β] [order_closed_topology β] {ι : Sort u_1} [nonempty ι] {f : α → β} {g : ι → α} (Cf : continuous_at f (infi g)) (Af : antitone f) :

f (⨅ (i : ι), g i) = ⨆ (i : ι), f (g i)

An antitone function continuous at the indexed infimum over a nonempty Sort sends this indexed infimum to the indexed supremum of the composition.

theorem antitone.map_infi_of_continuous_at {α : Type u} {β : Type v} [complete_linear_order α] [topological_space α] [order_topology α] [complete_linear_order β] [topological_space β] [order_closed_topology β] {ι : Sort u_1} {f : α → β} {g : ι → α} (Cf : continuous_at f (infi g)) (Af : antitone f) (ftop : f ⊤ = ⊥) :

f (infi g) = supr (f ∘ g)

If an antitone function sending top to bot is continuous at the indexed infimum over a Sort, then it sends this indexed infimum to the indexed supremum of the composition.

theorem cSup_mem_closure {α : Type u} [conditionally_complete_linear_order α] [topological_space α] [order_topology α] {s : set α} (hs : s.nonempty) (B : bdd_above s) :

has_Sup.Sup s ∈ closure s

theorem cInf_mem_closure {α : Type u} [conditionally_complete_linear_order α] [topological_space α] [order_topology α] {s : set α} (hs : s.nonempty) (B : bdd_below s) :

has_Inf.Inf s ∈ closure s

theorem is_closed.cSup_mem {α : Type u} [conditionally_complete_linear_order α] [topological_space α] [order_topology α] {s : set α} (hc : is_closed s) (hs : s.nonempty) (B : bdd_above s) :

has_Sup.Sup s ∈ s

theorem is_closed.cInf_mem {α : Type u} [conditionally_complete_linear_order α] [topological_space α] [order_topology α] {s : set α} (hc : is_closed s) (hs : s.nonempty) (B : bdd_below s) :

has_Inf.Inf s ∈ s

theorem monotone.map_cSup_of_continuous_at {α : Type u} {β : Type v} [conditionally_complete_linear_order α] [topological_space α] [order_topology α] [conditionally_complete_linear_order β] [topological_space β] [order_closed_topology β] {f : α → β} {s : set α} (Cf : continuous_at f (has_Sup.Sup s)) (Mf : monotone f) (ne : s.nonempty) (H : bdd_above s) :

f (has_Sup.Sup s) = has_Sup.Sup (f '' s)

If a monotone function is continuous at the supremum of a nonempty bounded above set s, then it sends this supremum to the supremum of the image of s.

theorem monotone.map_csupr_of_continuous_at {α : Type u} {β : Type v} {γ : Type w} [conditionally_complete_linear_order α] [topological_space α] [order_topology α] [conditionally_complete_linear_order β] [topological_space β] [order_closed_topology β] [nonempty γ] {f : α → β} {g : γ → α} (Cf : continuous_at f (⨆ (i : γ), g i)) (Mf : monotone f) (H : bdd_above (set.range g)) :

f (⨆ (i : γ), g i) = ⨆ (i : γ), f (g i)

If a monotone function is continuous at the indexed supremum of a bounded function on a nonempty Sort, then it sends this supremum to the supremum of the composition.

theorem monotone.map_cInf_of_continuous_at {α : Type u} {β : Type v} [conditionally_complete_linear_order α] [topological_space α] [order_topology α] [conditionally_complete_linear_order β] [topological_space β] [order_closed_topology β] {f : α → β} {s : set α} (Cf : continuous_at f (has_Inf.Inf s)) (Mf : monotone f) (ne : s.nonempty) (H : bdd_below s) :

f (has_Inf.Inf s) = has_Inf.Inf (f '' s)

If a monotone function is continuous at the infimum of a nonempty bounded below set s, then it sends this infimum to the infimum of the image of s.

theorem monotone.map_cinfi_of_continuous_at {α : Type u} {β : Type v} {γ : Type w} [conditionally_complete_linear_order α] [topological_space α] [order_topology α] [conditionally_complete_linear_order β] [topological_space β] [order_closed_topology β] [nonempty γ] {f : α → β} {g : γ → α} (Cf : continuous_at f (⨅ (i : γ), g i)) (Mf : monotone f) (H : bdd_below (set.range g)) :

f (⨅ (i : γ), g i) = ⨅ (i : γ), f (g i)

A continuous monotone function sends indexed infimum to indexed infimum in conditionally complete linear order, under a boundedness assumption.

theorem antitone.map_cSup_of_continuous_at {α : Type u} {β : Type v} [conditionally_complete_linear_order α] [topological_space α] [order_topology α] [conditionally_complete_linear_order β] [topological_space β] [order_closed_topology β] {f : α → β} {s : set α} (Cf : continuous_at f (has_Sup.Sup s)) (Af : antitone f) (ne : s.nonempty) (H : bdd_above s) :

f (has_Sup.Sup s) = has_Inf.Inf (f '' s)

If an antitone function is continuous at the supremum of a nonempty bounded above set s, then it sends this supremum to the infimum of the image of s.

theorem antitone.map_csupr_of_continuous_at {α : Type u} {β : Type v} {γ : Type w} [conditionally_complete_linear_order α] [topological_space α] [order_topology α] [conditionally_complete_linear_order β] [topological_space β] [order_closed_topology β] [nonempty γ] {f : α → β} {g : γ → α} (Cf : continuous_at f (⨆ (i : γ), g i)) (Af : antitone f) (H : bdd_above (set.range g)) :

f (⨆ (i : γ), g i) = ⨅ (i : γ), f (g i)

If an antitone function is continuous at the indexed supremum of a bounded function on a nonempty Sort, then it sends this supremum to the infimum of the composition.

theorem antitone.map_cInf_of_continuous_at {α : Type u} {β : Type v} [conditionally_complete_linear_order α] [topological_space α] [order_topology α] [conditionally_complete_linear_order β] [topological_space β] [order_closed_topology β] {f : α → β} {s : set α} (Cf : continuous_at f (has_Inf.Inf s)) (Af : antitone f) (ne : s.nonempty) (H : bdd_below s) :

f (has_Inf.Inf s) = has_Sup.Sup (f '' s)

If an antitone function is continuous at the infimum of a nonempty bounded below set s, then it sends this infimum to the supremum of the image of s.

theorem antitone.map_cinfi_of_continuous_at {α : Type u} {β : Type v} {γ : Type w} [conditionally_complete_linear_order α] [topological_space α] [order_topology α] [conditionally_complete_linear_order β] [topological_space β] [order_closed_topology β] [nonempty γ] {f : α → β} {g : γ → α} (Cf : continuous_at f (⨅ (i : γ), g i)) (Af : antitone f) (H : bdd_below (set.range g)) :

f (⨅ (i : γ), g i) = ⨆ (i : γ), f (g i)

A continuous antitone function sends indexed infimum to indexed supremum in conditionally complete linear order, under a boundedness assumption.

theorem monotone.tendsto_nhds_within_Iio {α : Type u_1} {β : Type u_2} [linear_order α] [topological_space α] [order_topology α] [conditionally_complete_linear_order β] [topological_space β] [order_topology β] {f : α → β} (Mf : monotone f) (x : α) :

filter.tendsto f (nhds_within x (set.Iio x)) (nhds (has_Sup.Sup (f '' set.Iio x)))

A monotone map has a limit to the left of any point x, equal to Sup (f '' (Iio x)).

theorem monotone.tendsto_nhds_within_Ioi {α : Type u_1} {β : Type u_2} [linear_order α] [topological_space α] [order_topology α] [conditionally_complete_linear_order β] [topological_space β] [order_topology β] {f : α → β} (Mf : monotone f) (x : α) :

filter.tendsto f (nhds_within x (set.Ioi x)) (nhds (has_Inf.Inf (f '' set.Ioi x)))

A monotone map has a limit to the right of any point x, equal to Inf (f '' (Ioi x)).

theorem eventually_nhds_within_pos_mem_Ioo {α : Type u} [linear_order α] [has_zero α] [topological_space α] [order_topology α] {ε : α} (h : 0 < ε) :

∀ᶠ (x : α) in nhds_within 0 (set.Ioi 0), x ∈ set.Ioo 0 ε

theorem eventually_nhds_within_pos_mem_Ioc {α : Type u} [linear_order α] [has_zero α] [topological_space α] [order_topology α] {ε : α} (h : 0 < ε) :

∀ᶠ (x : α) in nhds_within 0 (set.Ioi 0), x ∈ set.Ioc 0 ε