# mathlibdocumentation

topology.algebra.ordered.basic

# Theory of topology on ordered spaces #

## 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 all`x`.

### 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`.

### Connected sets and Intermediate Value Theorem #

• `is_preconnected_I??` : all intervals `I??` are preconnected,
• `is_preconnected.intermediate_value`, `intermediate_value_univ` : Intermediate Value Theorem for connected sets and connected spaces, respectively;
• `intermediate_value_Icc`, `intermediate_value_Icc'`: Intermediate Value Theorem for functions on closed intervals.

### Miscellaneous facts #

• `is_compact.exists_forall_le`, `is_compact.exists_forall_ge` : extreme value theorem, a continuous function on a compact set takes its minimum and maximum values.
• `is_closed.Icc_subset_of_forall_mem_nhds_within` : “Continuous induction” principle; if `s ∩ [a, b]` is closed, `a ∈ s`, and for each `x ∈ [a, b) ∩ s` some of its right neighborhoods is included `s`, then `[a, b] ⊆ s`.
• `is_closed.Icc_subset_of_forall_exists_gt`, `is_closed.mem_of_ge_of_forall_exists_gt` : two other versions of the “continuous induction” principle.

## Implementation #

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) [preorder α] :
Prop

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
@[instance]
def order_dual.topological_space {α : Type u}  :
Equations
@[instance]
@[instance]
@[instance]
def order_dual.has_continuous_mul {α : Type u} [has_mul α] [h : has_continuous_mul α] :
@[instance]
def subtype.order_closed_topology {α : Type u} [preorder α] [t : order_closed_topology α] {p : α → Prop} :
theorem is_closed_le_prod {α : Type u} [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} [preorder α] [t : order_closed_topology α] {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} [preorder α] [t : order_closed_topology α] (a : α) :
is_closed {b : α | b a}
theorem is_closed_Iic {α : Type u} [preorder α] [t : order_closed_topology α] {a : α} :
theorem is_closed_ge' {α : Type u} [preorder α] [t : order_closed_topology α] (a : α) :
is_closed {b : α | a b}
theorem is_closed_Ici {α : Type u} [preorder α] [t : order_closed_topology α] {a : α} :
@[instance]
theorem is_closed_Icc {α : Type u} [preorder α] [t : order_closed_topology α] {a b : α} :
@[simp]
theorem closure_Icc {α : Type u} [preorder α] [t : order_closed_topology α] (a b : α) :
closure (set.Icc a b) = b
@[simp]
theorem closure_Iic {α : Type u} [preorder α] [t : order_closed_topology α] (a : α) :
@[simp]
theorem closure_Ici {α : Type u} [preorder α] [t : order_closed_topology α] (a : α) :
theorem le_of_tendsto_of_tendsto {α : Type u} {β : Type v} [preorder α] [t : order_closed_topology α] {f g : β → α} {b : filter β} {a₁ a₂ : α} [b.ne_bot] (hf : (𝓝 a₁)) (hg : (𝓝 a₂)) (h : f ≤ᶠ[b] g) :
a₁ a₂
theorem le_of_tendsto_of_tendsto' {α : Type u} {β : Type v} [preorder α] [t : order_closed_topology α] {f g : β → α} {b : filter β} {a₁ a₂ : α} [b.ne_bot] (hf : (𝓝 a₁)) (hg : (𝓝 a₂)) (h : ∀ (x : β), f x g x) :
a₁ a₂
theorem le_of_tendsto {α : Type u} {β : Type v} [preorder α] [t : order_closed_topology α] {f : β → α} {a b : α} {x : filter β} [x.ne_bot] (lim : (𝓝 a)) (h : ∀ᶠ (c : β) in x, f c b) :
a b
theorem le_of_tendsto' {α : Type u} {β : Type v} [preorder α] [t : order_closed_topology α] {f : β → α} {a b : α} {x : filter β} [x.ne_bot] (lim : (𝓝 a)) (h : ∀ (c : β), f c b) :
a b
theorem ge_of_tendsto {α : Type u} {β : Type v} [preorder α] [t : order_closed_topology α] {f : β → α} {a b : α} {x : filter β} [x.ne_bot] (lim : (𝓝 a)) (h : ∀ᶠ (c : β) in x, b f c) :
b a
theorem ge_of_tendsto' {α : Type u} {β : Type v} [preorder α] [t : order_closed_topology α] {f : β → α} {a b : α} {x : filter β} [x.ne_bot] (lim : (𝓝 a)) (h : ∀ (c : β), b f c) :
b a
@[simp]
theorem closure_le_eq {α : Type u} {β : Type v} [preorder α] [t : order_closed_topology α] {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} [preorder α] [t : order_closed_topology α] {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} [preorder α] [t : order_closed_topology α] {f g : β → α} {s : set β} {x : β} (hx : x ) (hf : x) (hg : x) (h : ∀ (y : β), y sf y g y) :
f x g x
theorem is_closed.is_closed_le {α : Type u} {β : Type v} [preorder α] [t : order_closed_topology α] {f g : β → α} {s : set β} (hs : is_closed s) (hf : s) (hg : 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 nhds_within_Ici_ne_bot {α : Type u} [preorder α] {a b : α} (H₂ : a b) :
@[instance]
theorem nhds_within_Ici_self_ne_bot {α : Type u} [preorder α] (a : α) :
theorem nhds_within_Iic_ne_bot {α : Type u} [preorder α] {a b : α} (H : a b) :
@[instance]
theorem nhds_within_Iic_self_ne_bot {α : Type u} [preorder α] (a : α) :
@[instance]
theorem is_open_lt_prod {α : Type u} [linear_order α]  :
is_open {p : α × α | p.fst < p.snd}
theorem is_open_lt {α : Type u} {β : Type v} [linear_order α] {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} [linear_order α] {a : α} :
theorem is_open_Ioi {α : Type u} [linear_order α] {a : α} :
theorem is_open_Ioo {α : Type u} [linear_order α] {a b : α} :
@[simp]
theorem interior_Ioi {α : Type u} [linear_order α] {a : α} :
@[simp]
theorem interior_Iio {α : Type u} [linear_order α] {a : α} :
@[simp]
theorem interior_Ioo {α : Type u} [linear_order α] {a b : α} :
theorem eventually_le_of_tendsto_lt {α : Type u} {γ : Type w} [linear_order α] {l : filter γ} {f : γ → α} {u v : α} (hv : v < u) (h : (𝓝 v)) :
∀ᶠ (a : γ) in l, f a u
theorem eventually_ge_of_tendsto_gt {α : Type u} {γ : Type w} [linear_order α] {l : filter γ} {f : γ → α} {u v : α} (hv : u < v) (h : (𝓝 v)) :
∀ᶠ (a : γ) in l, u f a
theorem intermediate_value_univ₂ {α : Type u} {γ : Type w} [linear_order α] {a b : γ} {f g : γ → α} (hf : continuous f) (hg : continuous g) (ha : f a g a) (hb : g b f b) :
∃ (x : γ), f x = g x

Intermediate value theorem for two functions: if `f` and `g` are two continuous functions on a preconnected space and `f a ≤ g a` and `g b ≤ f b`, then for some `x` we have `f x = g x`.

theorem intermediate_value_univ₂_eventually₁ {α : Type u} {γ : Type w} [linear_order α] {a : γ} {l : filter γ} [l.ne_bot] {f g : γ → α} (hf : continuous f) (hg : continuous g) (ha : f a g a) (he : g ≤ᶠ[l] f) :
∃ (x : γ), f x = g x
theorem intermediate_value_univ₂_eventually₂ {α : Type u} {γ : Type w} [linear_order α] {l₁ l₂ : filter γ} [l₁.ne_bot] [l₂.ne_bot] {f g : γ → α} (hf : continuous f) (hg : continuous g) (he₁ : f ≤ᶠ[l₁] g) (he₂ : g ≤ᶠ[l₂] f) :
∃ (x : γ), f x = g x
theorem is_preconnected.intermediate_value₂ {α : Type u} {γ : Type w} [linear_order α] {s : set γ} (hs : is_preconnected s) {a b : γ} (ha : a s) (hb : b s) {f g : γ → α} (hf : s) (hg : s) (ha' : f a g a) (hb' : g b f b) :
∃ (x : γ) (H : x s), f x = g x

Intermediate value theorem for two functions: if `f` and `g` are two functions continuous on a preconnected set `s` and for some `a b ∈ s` we have `f a ≤ g a` and `g b ≤ f b`, then for some `x ∈ s` we have `f x = g x`.

theorem is_preconnected.intermediate_value₂_eventually₁ {α : Type u} {γ : Type w} [linear_order α] {s : set γ} (hs : is_preconnected s) {a : γ} {l : filter γ} (ha : a s) [l.ne_bot] (hl : l 𝓟 s) {f g : γ → α} (hf : s) (hg : s) (ha' : f a g a) (he : g ≤ᶠ[l] f) :
∃ (x : γ) (H : x s), f x = g x
theorem is_preconnected.intermediate_value₂_eventually₂ {α : Type u} {γ : Type w} [linear_order α] {s : set γ} (hs : is_preconnected s) {l₁ l₂ : filter γ} [l₁.ne_bot] [l₂.ne_bot] (hl₁ : l₁ 𝓟 s) (hl₂ : l₂ 𝓟 s) {f g : γ → α} (hf : s) (hg : s) (he₁ : f ≤ᶠ[l₁] g) (he₂ : g ≤ᶠ[l₂] f) :
∃ (x : γ) (H : x s), f x = g x
theorem is_preconnected.intermediate_value {α : Type u} {γ : Type w} [linear_order α] {s : set γ} (hs : is_preconnected s) {a b : γ} (ha : a s) (hb : b s) {f : γ → α} (hf : s) :
set.Icc (f a) (f b) f '' s

Intermediate Value Theorem for continuous functions on connected sets.

theorem is_preconnected.intermediate_value_Ico {α : Type u} {γ : Type w} [linear_order α] {s : set γ} (hs : is_preconnected s) {a : γ} {l : filter γ} (ha : a s) [l.ne_bot] (hl : l 𝓟 s) {f : γ → α} (hf : s) {v : α} (ht : (𝓝 v)) :
set.Ico (f a) v f '' s
theorem is_preconnected.intermediate_value_Ioc {α : Type u} {γ : Type w} [linear_order α] {s : set γ} (hs : is_preconnected s) {a : γ} {l : filter γ} (ha : a s) [l.ne_bot] (hl : l 𝓟 s) {f : γ → α} (hf : s) {v : α} (ht : (𝓝 v)) :
(f a) f '' s
theorem is_preconnected.intermediate_value_Ioo {α : Type u} {γ : Type w} [linear_order α] {s : set γ} (hs : is_preconnected s) {l₁ l₂ : filter γ} [l₁.ne_bot] [l₂.ne_bot] (hl₁ : l₁ 𝓟 s) (hl₂ : l₂ 𝓟 s) {f : γ → α} (hf : s) {v₁ v₂ : α} (ht₁ : l₁ (𝓝 v₁)) (ht₂ : l₂ (𝓝 v₂)) :
set.Ioo v₁ v₂ f '' s
theorem is_preconnected.intermediate_value_Ici {α : Type u} {γ : Type w} [linear_order α] {s : set γ} (hs : is_preconnected s) {a : γ} {l : filter γ} (ha : a s) [l.ne_bot] (hl : l 𝓟 s) {f : γ → α} (hf : s) (ht : filter.at_top) :
set.Ici (f a) f '' s
theorem is_preconnected.intermediate_value_Iic {α : Type u} {γ : Type w} [linear_order α] {s : set γ} (hs : is_preconnected s) {a : γ} {l : filter γ} (ha : a s) [l.ne_bot] (hl : l 𝓟 s) {f : γ → α} (hf : s) (ht : filter.at_bot) :
set.Iic (f a) f '' s
theorem is_preconnected.intermediate_value_Ioi {α : Type u} {γ : Type w} [linear_order α] {s : set γ} (hs : is_preconnected s) {l₁ l₂ : filter γ} [l₁.ne_bot] [l₂.ne_bot] (hl₁ : l₁ 𝓟 s) (hl₂ : l₂ 𝓟 s) {f : γ → α} (hf : s) {v : α} (ht₁ : l₁ (𝓝 v)) (ht₂ : l₂ filter.at_top) :
f '' s
theorem is_preconnected.intermediate_value_Iio {α : Type u} {γ : Type w} [linear_order α] {s : set γ} (hs : is_preconnected s) {l₁ l₂ : filter γ} [l₁.ne_bot] [l₂.ne_bot] (hl₁ : l₁ 𝓟 s) (hl₂ : l₂ 𝓟 s) {f : γ → α} (hf : s) {v : α} (ht₁ : l₁ filter.at_bot) (ht₂ : l₂ (𝓝 v)) :
f '' s
theorem is_preconnected.intermediate_value_Iii {α : Type u} {γ : Type w} [linear_order α] {s : set γ} (hs : is_preconnected s) {l₁ l₂ : filter γ} [l₁.ne_bot] [l₂.ne_bot] (hl₁ : l₁ 𝓟 s) (hl₂ : l₂ 𝓟 s) {f : γ → α} (hf : s) (ht₁ : l₁ filter.at_bot) (ht₂ : l₂ filter.at_top) :
theorem intermediate_value_univ {α : Type u} {γ : Type w} [linear_order α] (a b : γ) {f : γ → α} (hf : continuous f) :
set.Icc (f a) (f b)

Intermediate Value Theorem for continuous functions on connected spaces.

theorem mem_range_of_exists_le_of_exists_ge {α : Type u} {γ : Type w} [linear_order α] {c : α} {f : γ → α} (hf : continuous f) (h₁ : ∃ (a : γ), f a c) (h₂ : ∃ (b : γ), c f b) :
c

Intermediate Value Theorem for continuous functions on connected spaces.

theorem is_preconnected.Icc_subset {α : Type u} [linear_order α] {s : set α} (hs : is_preconnected s) {a b : α} (ha : a s) (hb : b s) :
b s

If a preconnected set contains endpoints of an interval, then it includes the whole interval.

theorem is_connected.Icc_subset {α : Type u} [linear_order α] {s : set α} (hs : is_connected s) {a b : α} (ha : a s) (hb : b s) :
b s

If a preconnected set contains endpoints of an interval, then it includes the whole interval.

theorem is_preconnected.eq_univ_of_unbounded {α : Type u} [linear_order α] {s : set α} (hs : is_preconnected s) (hb : ¬) (ha : ¬) :

If preconnected set in a linear order space is unbounded below and above, then it is the whole space.

### 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[Ioi a] a` and `𝓝[Ici a] 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} [linear_order α] {a b c : α} (H : b c) :
c 𝓝[] b
theorem Ioc_mem_nhds_within_Ioi {α : Type u} [linear_order α] {a b c : α} (H : b c) :
c 𝓝[] b
theorem Ico_mem_nhds_within_Ioi {α : Type u} [linear_order α] {a b c : α} (H : b c) :
c 𝓝[] b
theorem Icc_mem_nhds_within_Ioi {α : Type u} [linear_order α] {a b c : α} (H : b c) :
c 𝓝[] b
@[simp]
theorem nhds_within_Ioc_eq_nhds_within_Ioi {α : Type u} [linear_order α] {a b : α} (h : a < b) :
@[simp]
theorem nhds_within_Ioo_eq_nhds_within_Ioi {α : Type u} [linear_order α] {a b : α} (h : a < b) :
@[simp]
theorem continuous_within_at_Ioc_iff_Ioi {α : Type u} {β : Type v} [linear_order α] {a b : α} {f : α → β} (h : a < b) :
(set.Ioc a b) a a
@[simp]
theorem continuous_within_at_Ioo_iff_Ioi {α : Type u} {β : Type v} [linear_order α] {a b : α} {f : α → β} (h : a < b) :
(set.Ioo a b) a a

#### Left neighborhoods, point excluded #

theorem Ioo_mem_nhds_within_Iio {α : Type u} [linear_order α] {a b c : α} (H : b c) :
c 𝓝[] b
theorem Ico_mem_nhds_within_Iio {α : Type u} [linear_order α] {a b c : α} (H : b c) :
c 𝓝[] b
theorem Ioc_mem_nhds_within_Iio {α : Type u} [linear_order α] {a b c : α} (H : b c) :
c 𝓝[] b
theorem Icc_mem_nhds_within_Iio {α : Type u} [linear_order α] {a b c : α} (H : b c) :
c 𝓝[] b
@[simp]
theorem nhds_within_Ico_eq_nhds_within_Iio {α : Type u} [linear_order α] {a b : α} (h : a < b) :
@[simp]
theorem nhds_within_Ioo_eq_nhds_within_Iio {α : Type u} [linear_order α] {a b : α} (h : a < b) :
@[simp]
theorem continuous_within_at_Ico_iff_Iio {α : Type u} {γ : Type w} [linear_order α] {a b : α} {f : α → γ} (h : a < b) :
(set.Ico a b) b b
@[simp]
theorem continuous_within_at_Ioo_iff_Iio {α : Type u} {γ : Type w} [linear_order α] {a b : α} {f : α → γ} (h : a < b) :
(set.Ioo a b) b b

#### Right neighborhoods, point included #

theorem Ioo_mem_nhds_within_Ici {α : Type u} [linear_order α] {a b c : α} (H : b c) :
c 𝓝[] b
theorem Ioc_mem_nhds_within_Ici {α : Type u} [linear_order α] {a b c : α} (H : b c) :
c 𝓝[] b
theorem Ico_mem_nhds_within_Ici {α : Type u} [linear_order α] {a b c : α} (H : b c) :
c 𝓝[] b
theorem Icc_mem_nhds_within_Ici {α : Type u} [linear_order α] {a b c : α} (H : b c) :
c 𝓝[] b
@[simp]
theorem nhds_within_Icc_eq_nhds_within_Ici {α : Type u} [linear_order α] {a b : α} (h : a < b) :
@[simp]
theorem nhds_within_Ico_eq_nhds_within_Ici {α : Type u} [linear_order α] {a b : α} (h : a < b) :
@[simp]
theorem continuous_within_at_Icc_iff_Ici {α : Type u} {β : Type v} [linear_order α] {a b : α} {f : α → β} (h : a < b) :
(set.Icc a b) a a
@[simp]
theorem continuous_within_at_Ico_iff_Ici {α : Type u} {β : Type v} [linear_order α] {a b : α} {f : α → β} (h : a < b) :
(set.Ico a b) a a

#### Left neighborhoods, point included #

theorem Ioo_mem_nhds_within_Iic {α : Type u} [linear_order α] {a b c : α} (H : b c) :
c 𝓝[] b
theorem Ico_mem_nhds_within_Iic {α : Type u} [linear_order α] {a b c : α} (H : b c) :
c 𝓝[] b
theorem Ioc_mem_nhds_within_Iic {α : Type u} [linear_order α] {a b c : α} (H : b c) :
c 𝓝[] b
theorem Icc_mem_nhds_within_Iic {α : Type u} [linear_order α] {a b c : α} (H : b c) :
c 𝓝[] b
@[simp]
theorem nhds_within_Icc_eq_nhds_within_Iic {α : Type u} [linear_order α] {a b : α} (h : a < b) :
@[simp]
theorem nhds_within_Ioc_eq_nhds_within_Iic {α : Type u} [linear_order α] {a b : α} (h : a < b) :
@[simp]
theorem continuous_within_at_Icc_iff_Iic {α : Type u} {β : Type v} [linear_order α] {a b : α} {f : α → β} (h : a < b) :
(set.Icc a b) b b
@[simp]
theorem continuous_within_at_Ioc_iff_Iic {α : Type u} {β : Type v} [linear_order α] {a b : α} {f : α → β} (h : a < b) :
(set.Ioc a b) b b
theorem frontier_le_subset_eq {α : Type u} {β : Type v} [linear_order α] {f g : β → α} (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} [linear_order α] (a : α) :
theorem frontier_Ici_subset {α : Type u} [linear_order α] (a : α) :
theorem frontier_lt_subset_eq {α : Type u} {β : Type v} [linear_order α] {f g : β → α} (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} [linear_order α] {f g : β → α} [Π (x : β), decidable (f x g x)] {f' g' : β → γ} (hf : continuous f) (hg : continuous g) (hf' : {x : β | f x g x}) (hg' : {x : β | g x f x}) (hfg : ∀ (x : β), f x = g xf' 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} [linear_order α] {f g : β → α} [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 xf' x = g' x) :
continuous (λ (x : β), ite (f x g x) (f' x) (g' x))
theorem continuous.min {α : Type u} {β : Type v} [linear_order α] {f g : β → α} (hf : continuous f) (hg : continuous g) :
continuous (λ (b : β), min (f b) (g b))
theorem continuous.max {α : Type u} {β : Type v} [linear_order α] {f g : β → α} (hf : continuous f) (hg : continuous g) :
continuous (λ (b : β), max (f b) (g b))
theorem continuous_min {α : Type u} [linear_order α]  :
continuous (λ (p : α × α), min p.fst p.snd)
theorem continuous_max {α : Type u} [linear_order α]  :
continuous (λ (p : α × α), max p.fst p.snd)
theorem filter.tendsto.max {α : Type u} {β : Type v} [linear_order α] {f g : β → α} {b : filter β} {a₁ a₂ : α} (hf : (𝓝 a₁)) (hg : (𝓝 a₂)) :
filter.tendsto (λ (b : β), max (f b) (g b)) b (𝓝 (max a₁ a₂))
theorem filter.tendsto.min {α : Type u} {β : Type v} [linear_order α] {f g : β → α} {b : filter β} {a₁ a₂ : α} (hf : (𝓝 a₁)) (hg : (𝓝 a₂)) :
filter.tendsto (λ (b : β), min (f b) (g b)) b (𝓝 (min a₁ a₂))
theorem is_preconnected.ord_connected {α : Type u} [linear_order α] {s : set α} (h : is_preconnected s) :
@[instance]
def prod.order_closed_topology {α : Type u} {β : Type v} [preorder α] [preorder β]  :
@[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 (Π (i : ι), α i)
@[instance]
def pi.order_closed_topology' {α : Type u} {β : Type v} [preorder β]  :
@[class]
structure order_topology (α : Type u_1) [t : topological_space α] [preorder α] :
Prop

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
def preorder.topology (α : Type u_1) [preorder 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
@[instance]
def order_dual.order_topology {α : Type u_1}  :
theorem is_open_iff_generate_intervals {α : Type u} [t : order_topology α] {s : set α} :
topological_space.generate_open {s : set α | ∃ (a : α), s = s = set.Iio a} s
theorem is_open_lt' {α : Type u} [t : order_topology α] (a : α) :
is_open {b : α | a < b}
theorem is_open_gt' {α : Type u} [t : order_topology α] (a : α) :
is_open {b : α | b < a}
theorem lt_mem_nhds {α : Type u} [t : order_topology α] {a b : α} (h : a < b) :
∀ᶠ (x : α) in 𝓝 b, a < x
theorem le_mem_nhds {α : Type u} [t : order_topology α] {a b : α} (h : a < b) :
∀ᶠ (x : α) in 𝓝 b, a x
theorem gt_mem_nhds {α : Type u} [t : order_topology α] {a b : α} (h : a < b) :
∀ᶠ (x : α) in 𝓝 a, x < b
theorem ge_mem_nhds {α : Type u} [t : order_topology α] {a b : α} (h : a < b) :
∀ᶠ (x : α) in 𝓝 a, x b
theorem nhds_eq_order {α : Type u} [t : order_topology α] (a : α) :
𝓝 a = (⨅ (b : α) (H : b set.Iio a), 𝓟 (set.Ioi b)) ⨅ (b : α) (H : b set.Ioi a), 𝓟 (set.Iio b)
theorem tendsto_order {α : Type u} {β : Type v} [t : order_topology α] {f : β → α} {a : α} {x : filter β} :
(𝓝 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')
@[instance]
def tendsto_Icc_class_nhds {α : Type u} [t : order_topology α] (a : α) :
(𝓝 a)
@[instance]
def tendsto_Ico_class_nhds {α : Type u} [t : order_topology α] (a : α) :
(𝓝 a)
@[instance]
def tendsto_Ioc_class_nhds {α : Type u} [t : order_topology α] (a : α) :
(𝓝 a)
@[instance]
def tendsto_Ioo_class_nhds {α : Type u} [t : order_topology α] (a : α) :
(𝓝 a)
theorem tendsto_of_tendsto_of_tendsto_of_le_of_le' {α : Type u} {β : Type v} [t : order_topology α] {f g h : β → α} {b : filter β} {a : α} (hg : (𝓝 a)) (hh : (𝓝 a)) (hgf : ∀ᶠ (b : β) in b, g b f b) (hfh : ∀ᶠ (b : β) in b, f b h b) :
(𝓝 a)

Also known as squeeze or 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} [t : order_topology α] {f g h : β → α} {b : filter β} {a : α} (hg : (𝓝 a)) (hh : (𝓝 a)) (hgf : g f) (hfh : f h) :
(𝓝 a)

Also known as squeeze or sandwich theorem. This version assumes that inequalities hold everywhere.

theorem nhds_order_unbounded {α : Type u} [t : order_topology α] {a : α} (hu : ∃ (u : α), a < u) (hl : ∃ (l : α), l < a) :
𝓝 a = ⨅ (l : α) (h₂ : l < a) (u : α) (h₂ : a < u), 𝓟 (set.Ioo l u)
theorem tendsto_order_unbounded {α : Type u} {β : Type v} [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 < aa < u(∀ᶠ (b : β) in x, l < f b f b < u)) :
(𝓝 a)
@[instance]
def tendsto_Ixx_nhds_within {α : Type u_1} [preorder α] (a : α) {s t : set α} {Ixx : α → α → set α} [ (𝓝 a) (𝓝 a)] [ (𝓟 s) (𝓟 t)] :
(𝓝[s] a) (𝓝[t] a)
@[instance]
def tendsto_Icc_class_nhds_pi {ι : Type u_1} {α : ι → Type u_2} [Π (i : ι), partial_order (α i)] [Π (i : ι), topological_space (α i)] [∀ (i : ι), order_topology (α i)] (f : Π (i : ι), α i) :
(𝓝 f)
theorem induced_order_topology' {α : Type u} {β : Type v} [ta : topological_space 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)) :
theorem induced_order_topology {α : Type u} {β : Type v} [ta : topological_space 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)) :
@[instance]
def order_topology_of_ord_connected {α : Type u} [ta : topological_space α] [linear_order α] {t : set α} [ht : t.ord_connected] :

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_top_order {α : Type u} [order_top α]  :
= ⨅ (l : α) (h₂ : l < ), 𝓟 (set.Ioi l)
theorem nhds_bot_order {α : Type u} [order_bot α]  :
= ⨅ (l : α) (h₂ : < l), 𝓟 (set.Iio l)
theorem nhds_top_basis {α : Type u} [nontrivial α] :
(𝓝 ).has_basis (λ (a : α), a < ) (λ (a : α), set.Ioi a)
theorem nhds_bot_basis {α : Type u} [nontrivial α] :
(𝓝 ).has_basis (λ (a : α), < a) (λ (a : α), set.Iio a)
theorem nhds_top_basis_Ici {α : Type u} [nontrivial α]  :
(𝓝 ).has_basis (λ (a : α), a < ) set.Ici
theorem nhds_bot_basis_Iic {α : Type u} [nontrivial α]  :
(𝓝 ).has_basis (λ (a : α), < a) set.Iic
theorem tendsto_nhds_top_mono {α : Type u} {β : Type v} [order_top β] {l : filter α} {f g : α → β} (hf : (𝓝 )) (hg : f ≤ᶠ[l] g) :
theorem tendsto_nhds_bot_mono {α : Type u} {β : Type v} [order_bot β] {l : filter α} {f g : α → β} (hf : (𝓝 )) (hg : g ≤ᶠ[l] f) :
theorem tendsto_nhds_top_mono' {α : Type u} {β : Type v} [order_top β] {l : filter α} {f g : α → β} (hf : (𝓝 )) (hg : f g) :
theorem tendsto_nhds_bot_mono' {α : Type u} {β : Type v} [order_bot β] {l : filter α} {f g : α → β} (hf : (𝓝 )) (hg : g f) :
theorem exists_Ioc_subset_of_mem_nhds' {α : Type u} [linear_order α] {a : α} {s : set α} (hs : s 𝓝 a) {l : α} (hl : l < a) :
∃ (l' : α) (H : l' a), set.Ioc l' a s
theorem exists_Ico_subset_of_mem_nhds' {α : Type u} [linear_order α] {a : α} {s : set α} (hs : s 𝓝 a) {u : α} (hu : a < u) :
∃ (u' : α) (H : u' u), u' s
theorem exists_Ioc_subset_of_mem_nhds {α : Type u} [linear_order α] {a : α} {s : set α} (hs : s 𝓝 a) (h : ∃ (l : α), l < a) :
∃ (l : α) (H : l < a), a s
theorem exists_Ico_subset_of_mem_nhds {α : Type u} [linear_order α] {a : α} {s : set α} (hs : s 𝓝 a) (h : ∃ (u : α), a < u) :
∃ (u : α) (_x : a < u), u s
theorem order_separated {α : Type u} [linear_order α] {a₁ a₂ : α} (h : a₁ < a₂) :
∃ (u v : set α), a₁ u a₂ v ∀ (b₁ : α), b₁ u∀ (b₂ : α), b₂ vb₁ < b₂
@[instance]
theorem order_topology.t2_space {α : Type u} [linear_order α]  :
@[instance]
def order_topology.regular_space {α : Type u} [linear_order α]  :
theorem mem_nhds_iff_exists_Ioo_subset' {α : Type u} [linear_order α] {a : α} {s : set α} (hl : ∃ (l : α), l < a) (hu : ∃ (u : α), a < u) :
s 𝓝 a ∃ (l u : α), a u 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} [linear_order α] [no_top_order α] [no_bot_order α] {a : α} {s : set α} :
s 𝓝 a ∃ (l u : α), a u 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} [linear_order α] {a : α} (hl : ∃ (l : α), l < a) (hu : ∃ (u : α), a < u) :
(𝓝 a).has_basis (λ (b : α × α), b.fst < a a < b.snd) (λ (b : α × α), b.snd)
theorem nhds_basis_Ioo {α : Type u} [linear_order α] [no_top_order α] [no_bot_order α] (a : α) :
(𝓝 a).has_basis (λ (b : α × α), b.fst < a a < b.snd) (λ (b : α × α), b.snd)
theorem filter.eventually.exists_Ioo_subset {α : Type u} [linear_order α] [no_top_order α] [no_bot_order α] {a : α} {p : α → Prop} (hp : ∀ᶠ (x : α) in 𝓝 a, p x) :
∃ (l u : α), a u u {x : α | p x}
theorem Iio_mem_nhds {α : Type u} [linear_order α] {a b : α} (h : a < b) :
theorem Ioi_mem_nhds {α : Type u} [linear_order α] {a b : α} (h : a < b) :
theorem Iic_mem_nhds {α : Type u} [linear_order α] {a b : α} (h : a < b) :
theorem Ici_mem_nhds {α : Type u} [linear_order α] {a b : α} (h : a < b) :
theorem Ioo_mem_nhds {α : Type u} [linear_order α] {a b x : α} (ha : a < x) (hb : x < b) :
b 𝓝 x
theorem Ioc_mem_nhds {α : Type u} [linear_order α] {a b x : α} (ha : a < x) (hb : x < b) :
b 𝓝 x
theorem Ico_mem_nhds {α : Type u} [linear_order α] {a b x : α} (ha : a < x) (hb : x < b) :
b 𝓝 x
theorem Icc_mem_nhds {α : Type u} [linear_order α] {a b x : α} (ha : a < x) (hb : x < b) :
b 𝓝 x

### 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} [fintype 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) :
theorem pi_Iic_mem_nhds' {α : Type u} [linear_order α] {ι : Type u_1} [fintype ι] {a' x' : ι → α} (ha : ∀ (i : ι), x' i < a' i) :
theorem pi_Ici_mem_nhds {ι : Type u_1} {π : ι → Type u_2} [fintype 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) :
theorem pi_Ici_mem_nhds' {α : Type u} [linear_order α] {ι : Type u_1} [fintype ι] {a' x' : ι → α} (ha : ∀ (i : ι), a' i < x' i) :
theorem pi_Icc_mem_nhds {ι : Type u_1} {π : ι → Type u_2} [fintype 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) :
b 𝓝 x
theorem pi_Icc_mem_nhds' {α : Type u} [linear_order α] {ι : Type u_1} [fintype ι] {a' b' x' : ι → α} (ha : ∀ (i : ι), a' i < x' i) (hb : ∀ (i : ι), x' i < b' i) :
set.Icc a' b' 𝓝 x'
theorem pi_Iio_mem_nhds {ι : Type u_1} {π : ι → Type u_2} [fintype 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) :
theorem pi_Iio_mem_nhds' {α : Type u} [linear_order α] {ι : Type u_1} [fintype ι] {a' x' : ι → α} [nonempty ι] (ha : ∀ (i : ι), x' i < a' i) :
theorem pi_Ioi_mem_nhds {ι : Type u_1} {π : ι → Type u_2} [fintype 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) :
theorem pi_Ioi_mem_nhds' {α : Type u} [linear_order α] {ι : Type u_1} [fintype ι] {a' x' : ι → α} [nonempty ι] (ha : ∀ (i : ι), a' i < x' i) :
theorem pi_Ioc_mem_nhds {ι : Type u_1} {π : ι → Type u_2} [fintype 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) :
b 𝓝 x
theorem pi_Ioc_mem_nhds' {α : Type u} [linear_order α] {ι : Type u_1} [fintype 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' 𝓝 x'
theorem pi_Ico_mem_nhds {ι : Type u_1} {π : ι → Type u_2} [fintype 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) :
b 𝓝 x
theorem pi_Ico_mem_nhds' {α : Type u} [linear_order α] {ι : Type u_1} [fintype 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' 𝓝 x'
theorem pi_Ioo_mem_nhds {ι : Type u_1} {π : ι → Type u_2} [fintype 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) :
b 𝓝 x
theorem pi_Ioo_mem_nhds' {α : Type u} [linear_order α] {ι : Type u_1} [fintype 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' 𝓝 x'
theorem disjoint_nhds_at_top {α : Type u} [linear_order α] [no_top_order α] (x : α) :
@[simp]
theorem inf_nhds_at_top {α : Type u} [linear_order α] [no_top_order α] (x : α) :
theorem disjoint_nhds_at_bot {α : Type u} [linear_order α] [no_bot_order α] (x : α) :
@[simp]
theorem inf_nhds_at_bot {α : Type u} [linear_order α] [no_bot_order α] (x : α) :
theorem not_tendsto_nhds_of_tendsto_at_top {α : Type u} {β : Type v} [linear_order α] [no_top_order α] {F : filter β} [F.ne_bot] {f : β → α} (hf : filter.at_top) (x : α) :
¬ (𝓝 x)
theorem not_tendsto_at_top_of_tendsto_nhds {α : Type u} {β : Type v} [linear_order α] [no_top_order α] {F : filter β} [F.ne_bot] {f : β → α} {x : α} (hf : (𝓝 x)) :
theorem not_tendsto_nhds_of_tendsto_at_bot {α : Type u} {β : Type v} [linear_order α] [no_bot_order α] {F : filter β} [F.ne_bot] {f : β → α} (hf : filter.at_bot) (x : α) :
¬ (𝓝 x)
theorem not_tendsto_at_bot_of_tendsto_nhds {α : Type u} {β : Type v} [linear_order α] [no_bot_order α] {F : filter β} [F.ne_bot] {f : β → α} {x : α} (hf : (𝓝 x)) :

### 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} [linear_order α] {a b : α} (hab : a < b) (s : set α) :
[s 𝓝[] a, s 𝓝[ b] a, s 𝓝[ b] a, ∃ (u : α) (H : u b), u s, ∃ (u : α) (H : u set.Ioi a), u s].tfae

The following statements are equivalent:

1. `s` is a neighborhood of `a` within `(a, +∞)`
2. `s` is a neighborhood of `a` within `(a, b]`
3. `s` is a neighborhood of `a` within `(a, b)`
4. `s` includes `(a, u)` for some `u ∈ (a, b]`
5. `s` includes `(a, u)` for some `u > a`
theorem mem_nhds_within_Ioi_iff_exists_mem_Ioc_Ioo_subset {α : Type u} [linear_order α] {a u' : α} {s : set α} (hu' : a < u') :
s 𝓝[] a ∃ (u : α) (H : u u'), u s
theorem mem_nhds_within_Ioi_iff_exists_Ioo_subset' {α : Type u} [linear_order α] {a u' : α} {s : set α} (hu' : a < u') :
s 𝓝[] a ∃ (u : α) (H : u set.Ioi 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} [linear_order α] [no_top_order α] {a : α} {s : set α} :
s 𝓝[] a ∃ (u : α) (H : u set.Ioi 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} [linear_order α] [no_top_order α] {a : α} {s : set α} :
s 𝓝[] a ∃ (u : α) (H : u set.Ioi 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} [linear_order α] {a b : α} (h : a < b) (s : set α) :
[s 𝓝[] b, s 𝓝[ b] b, s 𝓝[ b] b, ∃ (l : α) (H : l b), b s, ∃ (l : α) (H : l set.Iio b), b s].tfae

The following statements are equivalent:

1. `s` is a neighborhood of `b` within `(-∞, b)`
2. `s` is a neighborhood of `b` within `[a, b)`
3. `s` is a neighborhood of `b` within `(a, b)`
4. `s` includes `(l, b)` for some `l ∈ [a, b)`
5. `s` includes `(l, b)` for some `l < b`
theorem mem_nhds_within_Iio_iff_exists_mem_Ico_Ioo_subset {α : Type u} [linear_order α] {a l' : α} {s : set α} (hl' : l' < a) :
s 𝓝[] a ∃ (l : α) (H : l set.Ico l' a), a s
theorem mem_nhds_within_Iio_iff_exists_Ioo_subset' {α : Type u} [linear_order α] {a l' : α} {s : set α} (hl' : l' < a) :
s 𝓝[] a ∃ (l : α) (H : l set.Iio a), 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} [linear_order α] [no_bot_order α] {a : α} {s : set α} :
s 𝓝[] a ∃ (l : α) (H : l set.Iio a), 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} [linear_order α] [no_bot_order α] {a : α} {s : set α} :
s 𝓝[] a ∃ (l : α) (H : l set.Iio a), 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} [linear_order α] {a b : α} (hab : a < b) (s : set α) :
[s 𝓝[] a, s 𝓝[ b] a, s 𝓝[ b] a, ∃ (u : α) (H : u b), u s, ∃ (u : α) (H : u set.Ioi a), u s].tfae

The following statements are equivalent:

1. `s` is a neighborhood of `a` within `[a, +∞)`
2. `s` is a neighborhood of `a` within `[a, b]`
3. `s` is a neighborhood of `a` within `[a, b)`
4. `s` includes `[a, u)` for some `u ∈ (a, b]`
5. `s` includes `[a, u)` for some `u > a`
theorem mem_nhds_within_Ici_iff_exists_mem_Ioc_Ico_subset {α : Type u} [linear_order α] {a u' : α} {s : set α} (hu' : a < u') :
s 𝓝[] a ∃ (u : α) (H : u u'), u s
theorem mem_nhds_within_Ici_iff_exists_Ico_subset' {α : Type u} [linear_order α] {a u' : α} {s : set α} (hu' : a < u') :
s 𝓝[] a ∃ (u : α) (H : u set.Ioi 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} [linear_order α] [no_top_order α] {a : α} {s : set α} :
s 𝓝[] a ∃ (u : α) (H : u set.Ioi 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_Ici_iff_exists_Icc_subset' {α : Type u} [linear_order α] [no_top_order α] {a : α} {s : set α} :
s 𝓝[] a ∃ (u : α) (H : u set.Ioi 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} [linear_order α] {a b : α} (h : a < b) (s : set α) :
[s 𝓝[] b, s 𝓝[ b] b, s 𝓝[ b] b, ∃ (l : α) (H : l b), b s, ∃ (l : α) (H : l set.Iio b), b s].tfae

The following statements are equivalent:

1. `s` is a neighborhood of `b` within `(-∞, b]`
2. `s` is a neighborhood of `b` within `[a, b]`
3. `s` is a neighborhood of `b` within `(a, b]`
4. `s` includes `(l, b]` for some `l ∈ [a, b)`
5. `s` includes `(l, b]` for some `l < b`
theorem mem_nhds_within_Iic_iff_exists_mem_Ico_Ioc_subset {α : Type u} [linear_order α] {a l' : α} {s : set α} (hl' : l' < a) :
s 𝓝[] a ∃ (l : α) (H : l set.Ico l' a), a s
theorem mem_nhds_within_Iic_iff_exists_Ioc_subset' {α : Type u} [linear_order α] {a l' : α} {s : set α} (hl' : l' < a) :
s 𝓝[] a ∃ (l : α) (H : l set.Iio a), 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} [linear_order α] [no_bot_order α] {a : α} {s : set α} :
s 𝓝[] a ∃ (l : α) (H : l set.Iio a), 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} [linear_order α] [no_bot_order α] {a : α} {s : set α} :
s 𝓝[] a ∃ (l : α) (H : l set.Iio a), 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_Ici_iff_exists_Icc_subset {α : Type u} [linear_order α] [no_top_order α] {a : α} {s : set α} :
s 𝓝[] a ∃ (u : α), a < u 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_Iic_iff_exists_Icc_subset {α : Type u} [linear_order α] [no_bot_order α] {a : α} {s : set α} :
s 𝓝[] a ∃ (l : α), l < a 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} (a : α) :
𝓝 a = ⨅ (r : α) (H : r > 0), 𝓟 {b : α | abs (a - b) < r}
theorem order_topology_of_nhds_abs {α : Type u_1} (h_nhds : ∀ (a : α), 𝓝 a = ⨅ (r : α) (H : r > 0), 𝓟 {b : α | abs (a - b) < r}) :
theorem linear_ordered_add_comm_group.tendsto_nhds {α : Type u} {β : Type v} {f : β → α} {x : filter β} {a : α} :
(𝓝 a) ∀ (ε : α), ε > 0(∀ᶠ (b : β) in x, abs (f b - a) < ε)
theorem eventually_abs_sub_lt {α : Type u} (a : α) {ε : α} (hε : 0 < ε) :
∀ᶠ (x : α) in 𝓝 a, abs (x - a) < ε
@[instance]
theorem continuous_abs {α : Type u}  :
theorem filter.tendsto.abs {α : Type u} {β : Type v} {f : β → α} {a : α} {l : filter β} (h : (𝓝 a)) :
filter.tendsto (λ (x : β), abs (f x)) l (𝓝 (abs a))
theorem nhds_basis_Ioo_pos {α : Type u} [no_bot_order α] [no_top_order α] (a : α) :
(𝓝 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} [no_bot_order α] [no_top_order α] (a : α) :
(𝓝 a).has_basis (λ (ε : α), 0 < ε) (λ (ε : α), {b : α | abs (b - a) < ε})
theorem nhds_basis_zero_abs_sub_lt (α : Type u) [no_bot_order α] [no_top_order α] :
(𝓝 0).has_basis (λ (ε : α), 0 < ε) (λ (ε : α), {b : α | abs b < ε})
theorem nhds_basis_Ioo_pos_of_pos {α : Type u} [no_bot_order α] [no_top_order α] {a : α} (ha : 0 < a) :
(𝓝 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 continuous.abs {α : Type u} {β : Type v} {f : β → α} (h : continuous f) :
continuous (λ (x : β), abs (f x))
theorem continuous_at.abs {α : Type u} {β : Type v} {f : β → α} {b : β} (h : b) :
continuous_at (λ (x : β), abs (f x)) b
theorem continuous_within_at.abs {α : Type u} {β : Type v} {f : β → α} {b : β} {s : set β} (h : b) :
continuous_within_at (λ (x : β), abs (f x)) s b
theorem continuous_on.abs {α : Type u} {β : Type v} {f : β → α} {s : set β} (h : s) :
continuous_on (λ (x : β), abs (f x)) s
theorem tendsto_abs_nhds_within_zero {α : Type u}  :
(𝓝[{0}] 0) (𝓝[] 0)
theorem filter.tendsto.add_at_top {α : Type u} {β : Type v} {l : filter β} {f g : β → α} {C : α} (hf : (𝓝 C)) (hg : 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} {l : filter β} {f g : β → α} {C : α} (hf : (𝓝 C)) (hg : 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} {l : filter β} {f g : β → α} {C : α} (hf : filter.at_top) (hg : (𝓝 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} {l : filter β} {f g : β → α} {C : α} (hf : filter.at_bot) (hg : (𝓝 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 mul_tendsto_nhds_zero_right {α : Type u} (x : α) :
(𝓝 0 ×ᶠ 𝓝 x) (𝓝 0)
theorem mul_tendsto_nhds_zero_left {α : Type u} (x : α) :
(𝓝 x ×ᶠ 𝓝 0) (𝓝 0)
theorem nhds_eq_map_mul_left_nhds_one {α : Type u} {x₀ : α} (hx₀ : x₀ 0) :
𝓝 x₀ = filter.map (λ (x : α), x₀ * x) (𝓝 1)
theorem nhds_eq_map_mul_right_nhds_one {α : Type u} {x₀ : α} (hx₀ : x₀ 0) :
𝓝 x₀ = filter.map (λ (x : α), x * x₀) (𝓝 1)
theorem mul_tendsto_nhds_one_nhds_one {α : Type u}  :
(𝓝 1 ×ᶠ 𝓝 1) (𝓝 1)
@[instance]
theorem filter.tendsto.at_top_mul {α : Type u} {β : Type v} {l : filter β} {f g : β → α} {C : α} (hC : 0 < C) (hf : filter.at_top) (hg : (𝓝 C)) :
filter.tendsto (λ (x : β), (f x) * g x) l filter.at_top

In a linearly ordered field with the order topology, if `f` tends to `at_top` and `g` tends to a positive constant `C` then `f * g` tends to `at_top`.

theorem filter.tendsto.mul_at_top {α : Type u} {β : Type v} {l : filter β} {f g : β → α} {C : α} (hC : 0 < C) (hf : (𝓝 C)) (hg : filter.at_top) :
filter.tendsto (λ (x : β), (f x) * g x) l filter.at_top

In a linearly ordered field with the order topology, if `f` tends to a positive constant `C` and `g` tends to `at_top` then `f * g` tends to `at_top`.

theorem filter.tendsto.at_top_mul_neg {α : Type u} {β : Type v} {l : filter β} {f g : β → α} {C : α} (hC : C < 0) (hf : filter.at_top) (hg : (𝓝 C)) :
filter.tendsto (λ (x : β), (f x) * g x) l filter.at_bot

In a linearly ordered field with the order topology, if `f` tends to `at_top` and `g` tends to a negative constant `C` then `f * g` tends to `at_bot`.

theorem filter.tendsto.neg_mul_at_top {α : Type u} {β : Type v} {l : filter β} {f g : β → α} {C : α} (hC : C < 0) (hf : (𝓝 C)) (hg : filter.at_top) :
filter.tendsto (λ (x : β), (f x) * g x) l filter.at_bot

In a linearly ordered field with the order topology, if `f` tends to a negative constant `C` and `g` tends to `at_top` then `f * g` tends to `at_bot`.

theorem filter.tendsto.at_bot_mul {α : Type u} {β : Type v} {l : filter β} {f g : β → α} {C : α} (hC : 0 < C) (hf : filter.at_bot) (hg : (𝓝 C)) :
filter.tendsto (λ (x : β), (f x) * g x) l filter.at_bot

In a linearly ordered field with the order topology, if `f` tends to `at_bot` and `g` tends to a positive constant `C` then `f * g` tends to `at_bot`.

theorem filter.tendsto.at_bot_mul_neg {α : Type u} {β : Type v} {l : filter β} {f g : β → α} {C : α} (hC : C < 0) (hf : filter.at_bot) (hg : (𝓝 C)) :
filter.tendsto (λ (x : β), (f x) * g x) l filter.at_top

In a linearly ordered field with the order topology, if `f` tends to `at_bot` and `g` tends to a negative constant `C` then `f * g` tends to `at_top`.

theorem filter.tendsto.mul_at_bot {α : Type u} {β : Type v} {l : filter β} {f g : β → α} {C : α} (hC : 0 < C) (hf : (𝓝 C)) (hg : filter.at_bot) :
filter.tendsto (λ (x : β), (f x) * g x) l filter.at_bot

In a linearly ordered field with the order topology, if `f` tends to a positive constant `C` and `g` tends to `at_bot` then `f * g` tends to `at_bot`.

theorem filter.tendsto.neg_mul_at_bot {α : Type u} {β : Type v} {l : filter β} {f g : β → α} {C : α} (hC : C < 0) (hf : (𝓝 C)) (hg : filter.at_bot) :
filter.tendsto (λ (x : β), (f x) * g x) l filter.at_top

In a linearly ordered field with the order topology, if `f` tends to a negative constant `C` and `g` tends to `at_bot` then `f * g` tends to `at_top`.

theorem tendsto_inv_zero_at_top {α : Type u}  :

The function `x ↦ x⁻¹` tends to `+∞` on the right of `0`.

theorem tendsto_inv_at_top_zero' {α : Type u}  :

The function `r ↦ r⁻¹` tends to `0` on the right as `r → +∞`.

theorem tendsto_inv_at_top_zero {α : Type u}  :
theorem filter.tendsto.div_at_top {α : Type u} {β : Type v} {f g : β → α} {l : filter β} {a : α} (h : (𝓝 a)) (hg : filter.at_top) :
filter.tendsto (λ (x : β), f x / g x) l (𝓝 0)
theorem filter.tendsto.inv_tendsto_at_top {α : Type u} {β : Type v} {l : filter β} {f : β → α} (h : filter.at_top) :
(𝓝 0)
theorem filter.tendsto.inv_tendsto_zero {α : Type u} {β : Type v} {l : filter β} {f : β → α} (h : (𝓝[] 0)) :
theorem tendsto_pow_neg_at_top {α : Type u} {n : } (hn : 1 n) :
filter.tendsto (λ (x : α), x ^ -n) filter.at_top (𝓝 0)

The function `x^(-n)` tends to `0` at `+∞` for any positive natural `n`. A version for positive real powers exists as `tendsto_rpow_neg_at_top`.

theorem tendsto_fpow_at_top_zero {α : Type u} {n : } (hn : n < 0) :
filter.tendsto (λ (x : α), x ^ n) filter.at_top (𝓝 0)
theorem tendsto_const_mul_fpow_at_top_zero {α : Type u} {n : } {c : α} (hn : n < 0) :
filter.tendsto (λ (x : α), c * x ^ n) filter.at_top (𝓝 0)
theorem tendsto_const_mul_pow_nhds_iff {α : Type u} {n : } {c d : α} (hc : c 0) :
filter.tendsto (λ (x : α), c * x ^ n) filter.at_top (𝓝 d) n = 0 c = d
theorem tendsto_const_mul_fpow_at_top_zero_iff {α : Type u} {n : } {c d : α} (hc : c 0) :
filter.tendsto (λ (x : α), c * x ^ n) filter.at_top (𝓝 d) n = 0 c = d n < 0 d = 0
theorem preimage_neg {α : Type u} [add_group α] :
theorem filter.map_neg {α : Type u} [add_group α] :
theorem is_lub.frequently_mem {α : Type u} [linear_order α] {a : α} {s : set α} (ha : a) (hs : s.nonempty) :
∃ᶠ (x : α) in 𝓝[] a, x s
theorem is_lub.frequently_nhds_mem {α : Type u} [linear_order α] {a : α} {s : set α} (ha : a) (hs : s.nonempty) :
∃ᶠ (x : α) in 𝓝 a, x s
theorem is_glb.frequently_mem {α : Type u} [linear_order α] {a : α} {s : set α} (ha : a) (hs : s.nonempty) :
∃ᶠ (x : α) in 𝓝[] a, x s
theorem is_glb.frequently_nhds_mem {α : Type u} [linear_order α] {a : α} {s : set α} (ha : a) (hs : s.nonempty) :
∃ᶠ (x : α) in 𝓝 a, x s
theorem is_lub.mem_closure {α : Type u} [linear_order α] {a : α} {s : set α} (ha : a) (hs : s.nonempty) :
a
theorem is_glb.mem_closure {α : Type u} [linear_order α] {a : α} {s : set α} (ha : a) (hs : s.nonempty) :
a
theorem is_lub.nhds_within_ne_bot {α : Type u} [linear_order α] {a : α} {s : set α} (ha : a) (hs : s.nonempty) :
theorem is_glb.nhds_within_ne_bot {α : Type u} [linear_order α] {a : α} {s : set α} :
as.nonempty(𝓝[s] a).ne_bot
theorem is_lub_of_mem_nhds {α : Type u} [linear_order α] {s : set α} {a : α} {f : filter α} (hsa : a ) (hsf : s f) [(f 𝓝 a).ne_bot] :
a
theorem is_lub_of_mem_closure {α : Type u} [linear_order α] {s : set α} {a : α} (hsa : a ) (hsf : a ) :
a
theorem is_glb_of_mem_nhds {α : Type u} [linear_order α] {s : set α} {a : α} {f : filter α} :
s f(f 𝓝 a).ne_bot a
theorem is_glb_of_mem_closure {α : Type u} [linear_order α] {s : set α} {a : α} (hsa : a ) (hsf : a ) :
a
theorem is_lub.mem_upper_bounds_of_tendsto {α : Type u} {γ : Type w} [linear_order α] [preorder γ] {f : α → γ} {s : set α} {a : α} {b : γ} (hf : ∀ (x : α), x s∀ (y : α), y sx yf x f y) (ha : a) (hb : (𝓝[s] a) (𝓝 b)) :
theorem is_lub.is_lub_of_tendsto {α : Type u} {γ : Type w} [linear_order α] [preorder γ] {f : α → γ} {s : set α} {a : α} {b : γ} (hf : ∀ (x : α), x s∀ (y : α), y sx yf x f y) (ha : a) (hs : s.nonempty) (hb : (𝓝[s] a) (𝓝 b)) :
is_lub (f '' s) b
theorem is_glb.mem_lower_bounds_of_tendsto {α : Type u} {γ : Type w} [linear_order α] [preorder γ] {f : α → γ} {s : set α} {a : α} {b : γ} (hf : ∀ (x : α), x s∀ (y : α), y sx yf x f y) (ha : a) (hb : (𝓝[s] a) (𝓝 b)) :
theorem is_glb.is_glb_of_tendsto {α : Type u} {γ : Type w} [linear_order α] [preorder γ] {f : α → γ} {s : set α} {a : α} {b : γ} (hf : ∀ (x : α), x s∀ (y : α), y sx yf x f y) :
as.nonempty (𝓝[s] a) (𝓝 b)is_glb (f '' s) b
theorem is_lub.mem_lower_bounds_of_tendsto {α : Type u} {γ : Type w} [linear_order α] [preorder γ] {f : α → γ} {s : set α} {a : α} {b : γ} (hf : ∀ (x : α), x s∀ (y : α), y sx yf y f x) (ha : a) (hb : (𝓝[s] a) (𝓝 b)) :
theorem is_lub.is_glb_of_tendsto {α : Type u} {γ : Type w} [linear_order α] [preorder γ] {f : α → γ} {s : set α} {a : α} {b : γ} :
(∀ (x : α), x s∀ (y : α), y sx yf y f x) as.nonempty (𝓝[s] a) (𝓝 b)is_glb (f '' s) b
theorem is_glb.mem_upper_bounds_of_tendsto {α : Type u} {γ : Type w} [linear_order α] [preorder γ] {f : α → γ} {s : set α} {a : α} {b : γ} (hf : ∀ (x : α), x s∀ (y : α), y sx yf y f x) (ha : a) (hb : (𝓝[s] a) (𝓝 b)) :
theorem is_glb.is_lub_of_tendsto {α : Type u} {γ : Type w} [linear_order α] [preorder γ] {f : α → γ} {s : set α} {a : α} {b : γ} :
(∀ (x : α), x s∀ (y : α), y sx yf y f x) as.nonempty (𝓝[s] a) (𝓝 b)is_lub (f '' s) b
theorem is_lub.mem_of_is_closed {α : Type u} [linear_order α] {a : α} {s : set α} (ha : a) (hs : s.nonempty) (sc : is_closed s) :
a s
theorem is_closed.is_lub_mem {α : Type u} [linear_order α] {a : α} {s : set α} (ha : 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} [linear_order α] {a : α} {s : set α} (ha : a) (hs : s.nonempty) (sc : is_closed s) :
a s
theorem is_closed.is_glb_mem {α : Type u} [linear_order α] {a : α} {s : set α} (ha : 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} [linear_order α] {t : set α} {x : α} (htx : x) (not_mem : x t) (ht : t.nonempty) (hx : (𝓝 x).is_countably_generated) :
∃ (u : → α), (∀ (n : ), u n < x) (𝓝 x) ∀ (n : ), u n t
theorem is_lub.exists_seq_monotone_tendsto' {α : Type u} [linear_order α] {t : set α} {x : α} (htx : x) (ht : t.nonempty) (hx : (𝓝 x).is_countably_generated) :
∃ (u : → α), (∀ (n : ), u n x) (𝓝 x) ∀ (n : ), u n t
theorem is_lub.exists_seq_strict_mono_tendsto_of_not_mem {α : Type u} [linear_order α] {t : set α} {x : α} (htx : x) (ht : t.nonempty) (not_mem : x t) :
∃ (u : → α), (∀ (n : ), u n < x) (𝓝 x) ∀ (n : ), u n t
theorem is_lub.exists_seq_monotone_tendsto {α : Type u} [linear_order α] {t : set α} {x : α} (htx : x) (ht : t.nonempty) :
∃ (u : → α), (∀ (n : ), u n x) (𝓝 x) ∀ (n : ), u n t
theorem exists_seq_strict_mono_tendsto' {α : Type u_1} [linear_order α] {x y : α} (hy : y < x) :
∃ (u : → α), (∀ (n : ), u n < x) (𝓝 x)
theorem exists_seq_strict_mono_tendsto {α : Type u} [linear_order α] [no_bot_order α] (x : α) :
∃ (u : → α), (∀ (n : ), u n < x) (𝓝 x)
theorem exists_seq_tendsto_Sup {α : Type u_1} {S : set α} (hS : S.nonempty) (hS' : bdd_above S) :
∃ (u : → α), (𝓝 (Sup S)) ∀ (n : ), u n S
theorem is_glb.exists_seq_strict_mono_tendsto_of_not_mem' {α : Type u} [linear_order α] {t : set α} {x : α} (htx : x) (not_mem : x t) (ht : t.nonempty) (hx : (𝓝 x).is_countably_generated) :
∃ (u : → α), (∀ (m n : ), m < nu n < u m) (∀ (n : ), x < u n) (𝓝 x) ∀ (n : ), u n t
theorem is_glb.exists_seq_monotone_tendsto' {α : Type u} [linear_order α] {t : set α} {x : α} (htx : x) (ht : t.nonempty) (hx : (𝓝 x).is_countably_generated) :
∃ (u : → α), (∀ (m n : ), m nu n u m) (∀ (n : ), x u n) (𝓝 x) ∀ (n : ), u n t
theorem is_glb.exists_seq_strict_mono_tendsto_of_not_mem {α : Type u} [linear_order α] {t : set α} {x : α} (htx : x) (ht : t.nonempty) (not_mem : x t) :
∃ (u : → α), (∀ (m n : ), m < nu n < u m) (∀ (n : ), x < u n) (𝓝 x) ∀ (n : ), u n t
theorem is_glb.exists_seq_monotone_tendsto {α : Type u} [linear_order α] {t : set α} {x : α} (htx : x) (ht : t.nonempty) :
∃ (u : → α), (∀ (m n : ), m nu n u m) (∀ (n : ), x u n) (𝓝 x) ∀ (n : ), u n t
theorem exists_seq_strict_antimono_tendsto' {α : Type u} [linear_order α] {x y : α} (hy : x < y) :
∃ (u : → α), (∀ (m n : ), m < nu n < u m) (∀ (n : ), x < u n) (𝓝 x)
theorem exists_seq_strict_antimono_tendsto {α : Type u} [linear_order α] [no_top_order α] (x : α) :
∃ (u : → α), (∀ (m n : ), m < nu n < u m) (∀ (n : ), x < u n) (𝓝 x)
theorem exists_seq_tendsto_Inf {α : Type u_1} {S : set α} (hS : S.nonempty) (hS' : bdd_below S) :
∃ (u : → α), (∀ (m n : ), m nu n u m) (𝓝 (Inf S)) ∀ (n : ), u n S
theorem is_compact.bdd_below {α : Type u} [linear_order α] [nonempty α] {s : set α} (hs : is_compact s) :

A compact set is bounded below

theorem is_compact.bdd_above {α : Type u} [linear_order α] [nonempty α] {s : set α} :

A compact set is bounded above

theorem closure_Ioi' {α : Type u} [linear_order α] {a b : α} (hab : a < b) :

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

@[simp]
theorem closure_Ioi {α : Type u} [linear_order α] (a : α) [no_top_order α] :

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

theorem closure_Iio' {α : Type u} [linear_order α] {a b : α} (hab : b < 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} [linear_order α] (a : α) [no_bot_order α] :

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

@[simp]
theorem closure_Ioo {α : Type u} [linear_order α] {a b : α} (hab : a < b) :
closure (set.Ioo a b) = b

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

@[simp]
theorem closure_Ioc {α : Type u} [linear_order α] {a b : α} (hab : a < b) :
closure (set.Ioc a b) = b

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

@[simp]
theorem closure_Ico {α : Type u} [linear_order α] {a b : α} (hab : a < b) :
closure (set.Ico a b) = b

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

@[simp]
theorem interior_Ici {α : Type u} [linear_order α] [no_bot_order α] {a : α} :
@[simp]
theorem interior_Iic {α : Type u} [linear_order α] [no_top_order α] {a : α} :
@[simp]
theorem interior_Icc {α : Type u} [linear_order α] [no_bot_order α] [no_top_order α] {a b : α} :
@[simp]
theorem interior_Ico {α : Type u} [linear_order α] [no_bot_order α] {a b : α} :
@[simp]
theorem interior_Ioc {α : Type u} [linear_order α] [no_top_order α] {a b : α} :
@[simp]
theorem frontier_Ici {α : Type u} [linear_order α] [no_bot_order α] {a : α} :
@[simp]
theorem frontier_Iic {α : Type u} [linear_order α] [no_top_order α] {a : α} :
@[simp]
theorem frontier_Ioi {α : Type u} [linear_order α] [no_top_order α] {a : α} :
@[simp]
theorem frontier_Iio {α : Type u} [linear_order α] [no_bot_order α] {a : α} :
@[simp]
theorem frontier_Icc {α : Type u} [linear_order α] [no_bot_order α] [no_top_order α] {a b : α} (h : a < b) :
frontier (set.Icc a b) = {a, b}
@[simp]
theorem frontier_Ioo {α : Type u} [linear_order α] {a b : α} (h : a < b) :
frontier (set.Ioo a b) = {a, b}
@[simp]
theorem frontier_Ico {α : Type u} [linear_order α] [no_bot_order α] {a b : α} (h : a < b) :
frontier (set.Ico a b) = {a, b}
@[simp]
theorem frontier_Ioc {α : Type u} [linear_order α] [no_top_order α] {a b : α} (h : a < b) :
frontier (set.Ioc a b) = {a, b}
theorem nhds_within_Ioi_ne_bot' {α : Type u} [linear_order α] {a b c : α} (H₁ : a < c) (H₂ : a b) :
theorem nhds_within_Ioi_ne_bot {α : Type u} [linear_order α] [no_top_order α]