Separation properties of topological spaces. #

This file defines the predicate separated, and common separation axioms (under the Kolmogorov classification).

Main definitions #

separated: Two sets are separated if they are contained in disjoint open sets.
t0_space: A T₀/Kolmogorov space is a space where, for every two points x ≠ y, there is an open set that contains one, but not the other.
t1_space: A T₁/Fréchet space is a space where every singleton set is closed. This is equivalent to, for every pair x ≠ y, there existing an open set containing x but not y (t1_space_iff_exists_open shows that these conditions are equivalent.)
t2_space: A T₂/Hausdorff space is a space where, for every two points x ≠ y, there is two disjoint open sets, one containing x, and the other y.
t2_5_space: A T₂.₅/Urysohn space is a space where, for every two points x ≠ y, there is two open sets, one containing x, and the other y, whose closures are disjoint.
regular_space: A T₃ space (sometimes referred to as regular, but authors vary on whether this includes T₂; mathlib does), is one where given any closed C and x ∉ C, there is disjoint open sets containing x and C respectively. In mathlib, T₃ implies T₂.₅.
normal_space: A T₄ space (sometimes referred to as normal, but authors vary on whether this includes T₂; mathlib does), is one where given two disjoint closed sets, we can find two open sets that separate them. In mathlib, T₄ implies T₃.

Main results #

T₀ spaces #

is_closed.exists_closed_singleton Given a closed set S in a compact T₀ space, there is some x ∈ S such that {x} is closed.
exists_open_singleton_of_open_finset Given an open finset S in a T₀ space, there is some x ∈ S such that {x} is open.

T₁ spaces #

is_closed_map_const: The constant map is a closed map.
discrete_of_t1_of_finite: A finite T₁ space must have the discrete topology.

T₂ spaces #

t2_iff_nhds: A space is T₂ iff the neighbourhoods of distinct points generate the bottom filter.
t2_iff_is_closed_diagonal: A space is T₂ iff the diagonal of α (that is, the set of all points of the form (a, a) : α × α) is closed under the product topology.
finset_disjoint_finset_opens_of_t2: Any two disjoint finsets are separated.
Most topological constructions preserve Hausdorffness; these results are part of the typeclass inference system (e.g. embedding.t2_space)
set.eq_on.closure: If two functions are equal on some set s, they are equal on its closure.
is_compact.is_closed: All compact sets are closed.
locally_compact_of_compact_nhds: If every point has a compact neighbourhood, then the space is locally compact.
tot_sep_of_zero_dim: If α has a clopen basis, it is a totally_separated_space.
loc_compact_t2_tot_disc_iff_tot_sep: A locally compact T₂ space is totally disconnected iff it is totally separated.

If the space is also compact:

normal_of_compact_t2: A compact T₂ space is a normal_space.
connected_components_eq_Inter_clopen: The connected component of a point is the intersection of all its clopen neighbourhoods.
compact_t2_tot_disc_iff_tot_sep: Being a totally_disconnected_space is equivalent to being a totally_separated_space.
connected_components.t2: connected_components α is T₂ for α T₂ and compact.

T₃ spaces #

disjoint_nested_nhds: Given two points x ≠ y, we can find neighbourhoods x ∈ V₁ ⊆ U₁ and y ∈ V₂ ⊆ U₂, with the Vₖ closed and the Uₖ open, such that the Uₖ are disjoint.

Discrete spaces #

discrete_topology_iff_nhds: Discrete topological spaces are those whose neighbourhood filters are the pure filter (which is the principal filter at a singleton).
induced_bot/discrete_topology_induced: The pullback of the discrete topology under an inclusion is the discrete topology.

References #

https://en.wikipedia.org/wiki/Separation_axiom

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def separated {α : Type u} [topological_space α] :

set α → set α → Prop

separated is a predicate on pairs of subsets of a topological space. It holds if the two subsets are contained in disjoint open sets.

Equations

separated = λ (s t : set α), ∃ (U V : set α), is_open U ∧ is_open V ∧ s ⊆ U ∧ t ⊆ V ∧ disjoint U V

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theorem separated.symm {α : Type u} [topological_space α] {s t : set α} :

separated s t → separated t s

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theorem separated.comm {α : Type u} [topological_space α] (s t : set α) :

separated s t ↔ separated t s

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theorem separated.empty_right {α : Type u} [topological_space α] (a : set α) :

separated a ∅

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theorem separated.empty_left {α : Type u} [topological_space α] (a : set α) :

separated ∅ a

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theorem separated.union_left {α : Type u} [topological_space α] {a b c : set α} :

separated a c → separated b c → separated (a ∪ b) c

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theorem separated.union_right {α : Type u} [topological_space α] {a b c : set α} (ab : separated a b) (ac : separated a c) :

separated a (b ∪ c)

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@[class]

structure t0_space (α : Type u) [topological_space α] :

Prop

t0 : ∀ (x y : α), x ≠ y → (∃ (U : set α), is_open U ∧ xor (x ∈ U) (y ∈ U))

A T₀ space, also known as a Kolmogorov space, is a topological space where for every pair x ≠ y, there is an open set containing one but not the other.

Instances

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theorem t0_space_def (α : Type u) [topological_space α] :

t0_space α ↔ ∀ (x y : α), x ≠ y → (∃ (U : set α), is_open U ∧ xor (x ∈ U) (y ∈ U))

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def indistinguishable {α : Type u} [topological_space α] (x y : α) :

Prop

Two points are topologically indistinguishable if no open set separates them.

Equations

indistinguishable x y = ∀ (U : set α), is_open U → (x ∈ U ↔ y ∈ U)

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theorem t0_space_iff_distinguishable (α : Type u) [topological_space α] :

t0_space α ↔ ∀ (x y : α), x ≠ y → ¬indistinguishable x y

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theorem indistinguishable_iff_closed {α : Type u} [topological_space α] (x y : α) :

indistinguishable x y ↔ ∀ (U : set α), is_closed U → (x ∈ U ↔ y ∈ U)

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theorem indistinguishable_iff_closure {α : Type u} [topological_space α] (x y : α) :

indistinguishable x y ↔ x ∈ closure {y} ∧ y ∈ closure {x}

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theorem subtype_indistinguishable_iff {α : Type u} [topological_space α] {U : set α} (x y : ↥U) :

indistinguishable x y ↔ indistinguishable ↑x ↑y

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theorem indistinguishable.eq {α : Type u} [topological_space α] [hα : t0_space α] {x y : α} (h : indistinguishable x y) :

x = y

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theorem is_closed.exists_closed_singleton {α : Type u_1} [topological_space α] [t0_space α] [compact_space α] {S : set α} (hS : is_closed S) (hne : S.nonempty) :

∃ (x : α), x ∈ S ∧ is_closed {x}

Given a closed set S in a compact T₀ space, there is some x ∈ S such that {x} is closed.

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theorem exists_open_singleton_of_open_finset {α : Type u} [topological_space α] [t0_space α] (s : finset α) (sne : s.nonempty) (hso : is_open ↑s) :

∃ (x : α) (H : x ∈ s), is_open {x}

Given an open finset S in a T₀ space, there is some x ∈ S such that {x} is open.

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theorem exists_open_singleton_of_fintype {α : Type u} [topological_space α] [t0_space α] [f : fintype α] [ha : nonempty α] :

∃ (x : α), is_open {x}

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@[protected, instance]

def subtype.t0_space {α : Type u} [topological_space α] [t0_space α] {p : α → Prop} :

t0_space (subtype p)

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theorem t0_space_iff_or_not_mem_closure (α : Type u) [topological_space α] :

t0_space α ↔ ∀ (a b : α), a ≠ b → a ∉ closure {b} ∨ b ∉ closure {a}

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theorem t0_space_of_injective_of_continuous {α β : Type u} [topological_space α] [topological_space β] {f : α → β} (hf : function.injective f) (hf' : continuous f) [t0_space β] :

t0_space α

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@[class]

structure t1_space (α : Type u) [topological_space α] :

Prop

t1 : ∀ (x : α), is_closed {x}

A T₁ space, also known as a Fréchet space, is a topological space where every singleton set is closed. Equivalently, for every pair x ≠ y, there is an open set containing x and not y.

Instances

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theorem is_closed_singleton {α : Type u} [topological_space α] [t1_space α] {x : α} :

is_closed {x}

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theorem is_open_compl_singleton {α : Type u} [topological_space α] [t1_space α] {x : α} :

is_open {x}ᶜ

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theorem is_open_ne {α : Type u} [topological_space α] [t1_space α] {x : α} :

is_open {y : α | y ≠ x}

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theorem ne.nhds_within_compl_singleton {α : Type u} [topological_space α] [t1_space α] {x y : α} (h : x ≠ y) :

𝓝[{y}ᶜ ] x = 𝓝 x

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theorem ne.nhds_within_diff_singleton {α : Type u} [topological_space α] [t1_space α] {x y : α} (h : x ≠ y) (s : set α) :

𝓝[s \ {y}] x = 𝓝[s] x

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@[protected]

theorem set.finite.is_closed {α : Type u} [topological_space α] [t1_space α] {s : set α} (hs : s.finite) :

is_closed s

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@[protected]

theorem finset.is_closed {α : Type u} [topological_space α] [t1_space α] (s : finset α) :

is_closed ↑s

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theorem t1_space_tfae (α : Type u) [t : topological_space α] :

[t1_space α, ∀ (x : α), is_closed {x}, ∀ (x : α), is_open {x}ᶜ, t ≤ cofinite_topology α, ∀ ⦃x y : α⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x, ∀ ⦃x y : α⦄, x ≠ y → (∃ (s : set α) (H : s ∈ 𝓝 x), y ∉ s), ∀ ⦃x y : α⦄, x ≠ y → (∃ (U : set α) (hU : is_open U), x ∈ U ∧ y ∉ U), ∀ ⦃x y : α⦄, x ≠ y → disjoint (𝓝 x) (pure y), ∀ ⦃x y : α⦄, x ≠ y → disjoint (pure x) (𝓝 y)].tfae

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theorem t1_space_iff_le_cofinite {α : Type u_1} [t : topological_space α] :

t1_space α ↔ t ≤ cofinite_topology α

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theorem t1_space_iff_exists_open {α : Type u} [topological_space α] :

t1_space α ↔ ∀ (x y : α), x ≠ y → (∃ (U : set α) (hU : is_open U), x ∈ U ∧ y ∉ U)

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theorem t1_space_iff_disjoint_pure_nhds {α : Type u} [topological_space α] :

t1_space α ↔ ∀ ⦃x y : α⦄, x ≠ y → disjoint (pure x) (𝓝 y)

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theorem t1_space_iff_disjoint_nhds_pure {α : Type u} [topological_space α] :

t1_space α ↔ ∀ ⦃x y : α⦄, x ≠ y → disjoint (𝓝 x) (pure y)

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theorem disjoint_pure_nhds {α : Type u} [topological_space α] [t1_space α] {x y : α} (h : x ≠ y) :

disjoint (pure x) (𝓝 y)

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theorem disjoint_nhds_pure {α : Type u} [topological_space α] [t1_space α] {x y : α} (h : x ≠ y) :

disjoint (𝓝 x) (pure y)

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@[protected, instance]

def t1_space_cofinite {α : Type u_1} :

t1_space α

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theorem t1_space_antitone {α : Type u_1} :

antitone (t1_space α)

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theorem continuous_within_at_update_of_ne {α : Type u} {β : Type v} [topological_space α] [t1_space α] [decidable_eq α] [topological_space β] {f : α → β} {s : set α} {x y : α} {z : β} (hne : y ≠ x) :

continuous_within_at (function.update f x z) s y ↔ continuous_within_at f s y

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theorem continuous_at_update_of_ne {α : Type u} {β : Type v} [topological_space α] [t1_space α] [decidable_eq α] [topological_space β] {f : α → β} {x y : α} {z : β} (hne : y ≠ x) :

continuous_at (function.update f x z) y ↔ continuous_at f y

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theorem continuous_on_update_iff {α : Type u} {β : Type v} [topological_space α] [t1_space α] [decidable_eq α] [topological_space β] {f : α → β} {s : set α} {x : α} {y : β} :

continuous_on (function.update f x y) s ↔ continuous_on f (s \ {x}) ∧ (x ∈ s → filter.tendsto f (𝓝[s \ {x}] x) (𝓝 y))

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@[protected, instance]

def subtype.t1_space {α : Type u} [topological_space α] [t1_space α] {p : α → Prop} :

t1_space (subtype p)

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@[protected, instance]

def t1_space.t0_space {α : Type u} [topological_space α] [t1_space α] :

t0_space α

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theorem compl_singleton_mem_nhds {α : Type u} [topological_space α] [t1_space α] {x y : α} (h : y ≠ x) :

{x}ᶜ ∈ 𝓝 y

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@[simp]

theorem closure_singleton {α : Type u} [topological_space α] [t1_space α] {a : α} :

closure {a} = {a}

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theorem set.subsingleton.closure {α : Type u} [topological_space α] [t1_space α] {s : set α} (hs : s.subsingleton) :

(closure s).subsingleton

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@[simp]

theorem subsingleton_closure {α : Type u} [topological_space α] [t1_space α] {s : set α} :

(closure s).subsingleton ↔ s.subsingleton

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theorem is_closed_map_const {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] [t1_space β] {y : β} :

is_closed_map (function.const α y)

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theorem bInter_basis_nhds {α : Type u} [topological_space α] [t1_space α] {ι : Sort u_1} {p : ι → Prop} {s : ι → set α} {x : α} (h : (𝓝 x).has_basis p s) :

(⋂ (i : ι) (h : p i), s i) = {x}

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@[simp]

theorem nhds_le_nhds_iff {α : Type u} [topological_space α] [t1_space α] {a b : α} :

𝓝 a ≤ 𝓝 b ↔ a = b

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@[simp]

theorem nhds_eq_nhds_iff {α : Type u} [topological_space α] [t1_space α] {a b : α} :

𝓝 a = 𝓝 b ↔ a = b

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theorem dense.diff_singleton {α : Type u} [topological_space α] [t1_space α] {s : set α} (hs : dense s) (x : α) [(𝓝[{x}ᶜ ] x).ne_bot] :

dense (s \ {x})

Removing a non-isolated point from a dense set, one still obtains a dense set.

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theorem dense.diff_finset {α : Type u} [topological_space α] [t1_space α] [∀ (x : α), (𝓝[{x}ᶜ ] x).ne_bot] {s : set α} (hs : dense s) (t : finset α) :

dense (s \ ↑t)

Removing a finset from a dense set in a space without isolated points, one still obtains a dense set.

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theorem dense.diff_finite {α : Type u} [topological_space α] [t1_space α] [∀ (x : α), (𝓝[{x}ᶜ ] x).ne_bot] {s : set α} (hs : dense s) {t : set α} (ht : t.finite) :

dense (s \ t)

Removing a finite set from a dense set in a space without isolated points, one still obtains a dense set.

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theorem eq_of_tendsto_nhds {α : Type u} {β : Type v} [topological_space α] [topological_space β] [t1_space β] {f : α → β} {a : α} {b : β} (h : filter.tendsto f (𝓝 a) (𝓝 b)) :

f a = b

If a function to a t1_space tends to some limit b at some point a, then necessarily b = f a.

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theorem continuous_at_of_tendsto_nhds {α : Type u} {β : Type v} [topological_space α] [topological_space β] [t1_space β] {f : α → β} {a : α} {b : β} (h : filter.tendsto f (𝓝 a) (𝓝 b)) :

continuous_at f a

To prove a function to a t1_space is continuous at some point a, it suffices to prove that f admits some limit at a.

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theorem infinite_of_mem_nhds {α : Type u_1} [topological_space α] [t1_space α] (x : α) [hx : (𝓝[{x}ᶜ ] x).ne_bot] {s : set α} (hs : s ∈ 𝓝 x) :

s.infinite

If the punctured neighborhoods of a point form a nontrivial filter, then any neighborhood is infinite.

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theorem discrete_of_t1_of_finite {X : Type u_1} [topological_space X] [t1_space X] [fintype X] :

discrete_topology X

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theorem singleton_mem_nhds_within_of_mem_discrete {α : Type u} [topological_space α] {s : set α} [discrete_topology ↥s] {x : α} (hx : x ∈ s) :

{x} ∈ 𝓝[s] x

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theorem nhds_within_of_mem_discrete {α : Type u} [topological_space α] {s : set α} [discrete_topology ↥s] {x : α} (hx : x ∈ s) :

𝓝[s] x = pure x

The neighbourhoods filter of x within s, under the discrete topology, is equal to the pure x filter (which is the principal filter at the singleton {x}.)

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theorem filter.has_basis.exists_inter_eq_singleton_of_mem_discrete {α : Type u} [topological_space α] {ι : Type u_1} {p : ι → Prop} {t : ι → set α} {s : set α} [discrete_topology ↥s] {x : α} (hb : (𝓝 x).has_basis p t) (hx : x ∈ s) :

∃ (i : ι) (hi : p i), t i ∩ s = {x}

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theorem nhds_inter_eq_singleton_of_mem_discrete {α : Type u} [topological_space α] {s : set α} [discrete_topology ↥s] {x : α} (hx : x ∈ s) :

∃ (U : set α) (H : U ∈ 𝓝 x), U ∩ s = {x}

A point x in a discrete subset s of a topological space admits a neighbourhood that only meets s at x.

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theorem disjoint_nhds_within_of_mem_discrete {α : Type u} [topological_space α] {s : set α} [discrete_topology ↥s] {x : α} (hx : x ∈ s) :

∃ (U : set α) (H : U ∈ 𝓝[{x}ᶜ ] x), disjoint U s

For point x in a discrete subset s of a topological space, there is a set U such that

U is a punctured neighborhood of x (ie. U ∪ {x} is a neighbourhood of x),
U is disjoint from s.

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theorem topological_space.subset_trans {X : Type u_1} [tX : topological_space X] {s t : set X} (ts : t ⊆ s) :

subtype.topological_space = topological_space.induced (set.inclusion ts) subtype.topological_space

Let X be a topological space and let s, t ⊆ X be two subsets. If there is an inclusion t ⊆ s, then the topological space structure on t induced by X is the same as the one obtained by the induced topological space structure on s.

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theorem discrete_topology_iff_nhds {X : Type u_1} [topological_space X] :

discrete_topology X ↔ 𝓝 = pure

This lemma characterizes discrete topological spaces as those whose singletons are neighbourhoods.

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theorem induced_bot {X : Type u_1} {Y : Type u_2} {f : X → Y} (hf : function.injective f) :

topological_space.induced f ⊥ = ⊥

The topology pulled-back under an inclusion f : X → Y from the discrete topology (⊥) is the discrete topology. This version does not assume the choice of a topology on either the source X nor the target Y of the inclusion f.

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theorem discrete_topology_induced {X : Type u_1} {Y : Type u_2} [tY : topological_space Y] [discrete_topology Y] {f : X → Y} (hf : function.injective f) :

discrete_topology X

The topology induced under an inclusion f : X → Y from the discrete topological space Y is the discrete topology on X.

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theorem discrete_topology.of_subset {X : Type u_1} [topological_space X] {s t : set X} (ds : discrete_topology ↥s) (ts : t ⊆ s) :

discrete_topology ↥t

Let s, t ⊆ X be two subsets of a topological space X. If t ⊆ s and the topology induced by Xon s is discrete, then also the topology induces on t is discrete.

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@[class]

structure t2_space (α : Type u) [topological_space α] :

Prop

t2 : ∀ (x y : α), x ≠ y → (∃ (u v : set α), is_open u ∧ is_open v ∧ x ∈ u ∧ y ∈ v ∧ u ∩ v = ∅)

A T₂ space, also known as a Hausdorff space, is one in which for every x ≠ y there exists disjoint open sets around x and y. This is the most widely used of the separation axioms.

Instances

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theorem t2_separation {α : Type u} [topological_space α] [t2_space α] {x y : α} (h : x ≠ y) :

∃ (u v : set α), is_open u ∧ is_open v ∧ x ∈ u ∧ y ∈ v ∧ u ∩ v = ∅

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@[protected, instance]

def t2_space.t1_space {α : Type u} [topological_space α] [t2_space α] :

t1_space α

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theorem eq_of_nhds_ne_bot {α : Type u} [topological_space α] [ht : t2_space α] {x y : α} (h : (𝓝 x ⊓ 𝓝 y).ne_bot) :

x = y

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theorem t2_iff_nhds {α : Type u} [topological_space α] :

t2_space α ↔ ∀ {x y : α}, (𝓝 x ⊓ 𝓝 y).ne_bot → x = y

A space is T₂ iff the neighbourhoods of distinct points generate the bottom filter.

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theorem t2_iff_ultrafilter {α : Type u} [topological_space α] :

t2_space α ↔ ∀ {x y : α} (f : ultrafilter α), ↑f ≤ 𝓝 x → ↑f ≤ 𝓝 y → x = y

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theorem is_closed_diagonal {α : Type u} [topological_space α] [t2_space α] :

is_closed (set.diagonal α)

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theorem t2_iff_is_closed_diagonal {α : Type u} [topological_space α] :

t2_space α ↔ is_closed (set.diagonal α)

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theorem finset_disjoint_finset_opens_of_t2 {α : Type u} [topological_space α] [t2_space α] (s t : finset α) :

disjoint s t → separated ↑s ↑t

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theorem point_disjoint_finset_opens_of_t2 {α : Type u} [topological_space α] [t2_space α] {x : α} {s : finset α} (h : x ∉ s) :

separated {x} ↑s

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theorem tendsto_nhds_unique {α : Type u} {β : Type v} [topological_space α] [t2_space α] {f : β → α} {l : filter β} {a b : α} [l.ne_bot] (ha : filter.tendsto f l (𝓝 a)) (hb : filter.tendsto f l (𝓝 b)) :

a = b

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theorem tendsto_nhds_unique' {α : Type u} {β : Type v} [topological_space α] [t2_space α] {f : β → α} {l : filter β} {a b : α} (hl : l.ne_bot) (ha : filter.tendsto f l (𝓝 a)) (hb : filter.tendsto f l (𝓝 b)) :

a = b

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theorem tendsto_nhds_unique_of_eventually_eq {α : Type u} {β : Type v} [topological_space α] [t2_space α] {f g : β → α} {l : filter β} {a b : α} [l.ne_bot] (ha : filter.tendsto f l (𝓝 a)) (hb : filter.tendsto g l (𝓝 b)) (hfg : f =ᶠ[l] g) :

a = b

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theorem tendsto_nhds_unique_of_frequently_eq {α : Type u} {β : Type v} [topological_space α] [t2_space α] {f g : β → α} {l : filter β} {a b : α} (ha : filter.tendsto f l (𝓝 a)) (hb : filter.tendsto g l (𝓝 b)) (hfg : ∃ᶠ (x : β) in l, f x = g x) :

a = b

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theorem tendsto_const_nhds_iff {α : Type u} [topological_space α] [t2_space α] {l : filter α} [l.ne_bot] {c d : α} :

filter.tendsto (λ (x : α), c) l (𝓝 d) ↔ c = d

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@[class]

structure t2_5_space (α : Type u) [topological_space α] :

Prop

t2_5 : ∀ (x y : α), x ≠ y → (∃ (U V : set α), is_open U ∧ is_open V ∧ closure U ∩ closure V = ∅ ∧ x ∈ U ∧ y ∈ V)

A T₂.₅ space, also known as a Urysohn space, is a topological space where for every pair x ≠ y, there are two open sets, with the intersection of closures empty, one containing x and the other y .

Instances

regular_space.t2_5_space

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@[protected, instance]

def t2_5_space.t2_space {α : Type u} [topological_space α] [t2_5_space α] :

t2_space α

Properties of `Lim` and `lim` #

In this section we use explicit nonempty α instances for Lim and lim. This way the lemmas are useful without a nonempty α instance.

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theorem Lim_eq {α : Type u} [topological_space α] [t2_space α] {f : filter α} {a : α} [f.ne_bot] (h : f ≤ 𝓝 a) :

Lim f = a

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theorem Lim_eq_iff {α : Type u} [topological_space α] [t2_space α] {f : filter α} [f.ne_bot] (h : ∃ (a : α), f ≤ 𝓝 a) {a : α} :

Lim f = a ↔ f ≤ 𝓝 a

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theorem ultrafilter.Lim_eq_iff_le_nhds {α : Type u} [topological_space α] [t2_space α] [compact_space α] {x : α} {F : ultrafilter α} :

F.Lim = x ↔ ↑F ≤ 𝓝 x

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theorem is_open_iff_ultrafilter' {α : Type u} [topological_space α] [t2_space α] [compact_space α] (U : set α) :

is_open U ↔ ∀ (F : ultrafilter α), F.Lim ∈ U → U ∈ F.to_filter

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theorem filter.tendsto.lim_eq {α : Type u} {β : Type v} [topological_space α] [t2_space α] {a : α} {f : filter β} [f.ne_bot] {g : β → α} (h : filter.tendsto g f (𝓝 a)) :

lim f g = a

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theorem filter.lim_eq_iff {α : Type u} {β : Type v} [topological_space α] [t2_space α] {f : filter β} [f.ne_bot] {g : β → α} (h : ∃ (a : α), filter.tendsto g f (𝓝 a)) {a : α} :

lim f g = a ↔ filter.tendsto g f (𝓝 a)

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theorem continuous.lim_eq {α : Type u} {β : Type v} [topological_space α] [t2_space α] [topological_space β] {f : β → α} (h : continuous f) (a : β) :

lim (𝓝 a) f = f a

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@[simp]

theorem Lim_nhds {α : Type u} [topological_space α] [t2_space α] (a : α) :

Lim (𝓝 a) = a

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@[simp]

theorem lim_nhds_id {α : Type u} [topological_space α] [t2_space α] (a : α) :

lim (𝓝 a) id = a

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@[simp]

theorem Lim_nhds_within {α : Type u} [topological_space α] [t2_space α] {a : α} {s : set α} (h : a ∈ closure s) :

Lim (𝓝[s] a) = a

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@[simp]

theorem lim_nhds_within_id {α : Type u} [topological_space α] [t2_space α] {a : α} {s : set α} (h : a ∈ closure s) :

lim (𝓝[s] a) id = a

`t2_space` constructions #

We use two lemmas to prove that various standard constructions generate Hausdorff spaces from Hausdorff spaces:

separated_by_continuous says that two points x y : α can be separated by open neighborhoods provided that there exists a continuous map f : α → β with a Hausdorff codomain such that f x ≠ f y. We use this lemma to prove that topological spaces defined using induced are Hausdorff spaces.
separated_by_open_embedding says that for an open embedding f : α → β of a Hausdorff space α, the images of two distinct points x y : α, x ≠ y can be separated by open neighborhoods. We use this lemma to prove that topological spaces defined using coinduced are Hausdorff spaces.

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@[protected, instance]

def t2_space_discrete {α : Type u_1} [topological_space α] [discrete_topology α] :

t2_space α

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theorem separated_by_continuous {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] [t2_space β] {f : α → β} (hf : continuous f) {x y : α} (h : f x ≠ f y) :

∃ (u v : set α), is_open u ∧ is_open v ∧ x ∈ u ∧ y ∈ v ∧ u ∩ v = ∅

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theorem separated_by_open_embedding {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] [t2_space α] {f : α → β} (hf : open_embedding f) {x y : α} (h : x ≠ y) :

∃ (u v : set β), is_open u ∧ is_open v ∧ f x ∈ u ∧ f y ∈ v ∧ u ∩ v = ∅

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@[protected, instance]

def subtype.t2_space {α : Type u_1} {p : α → Prop} [t : topological_space α] [t2_space α] :

t2_space (subtype p)

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@[protected, instance]

def prod.t2_space {α : Type u_1} {β : Type u_2} [t₁ : topological_space α] [t2_space α] [t₂ : topological_space β] [t2_space β] :

t2_space (α × β)

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theorem embedding.t2_space {α : Type u} {β : Type v} [topological_space α] [topological_space β] [t2_space β] {f : α → β} (hf : embedding f) :

t2_space α

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@[protected, instance]

def sum.t2_space {α : Type u_1} {β : Type u_2} [t₁ : topological_space α] [t2_space α] [t₂ : topological_space β] [t2_space β] :

t2_space (α ⊕ β)

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@[protected, instance]

def Pi.t2_space {α : Type u_1} {β : α → Type v} [t₂ : Π (a : α), topological_space (β a)] [∀ (a : α), t2_space (β a)] :

t2_space (Π (a : α), β a)

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@[protected, instance]

def sigma.t2_space {ι : Type u_1} {α : ι → Type u_2} [Π (i : ι), topological_space (α i)] [∀ (a : ι), t2_space (α a)] :

t2_space (Σ (i : ι), α i)

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theorem is_closed_eq {α : Type u} {β : Type v} [topological_space α] [topological_space β] [t2_space α] {f g : β → α} (hf : continuous f) (hg : continuous g) :

is_closed {x : β | f x = g x}

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theorem set.eq_on.closure {α : Type u} {β : Type v} [topological_space α] [topological_space β] [t2_space α] {s : set β} {f g : β → α} (h : set.eq_on f g s) (hf : continuous f) (hg : continuous g) :

set.eq_on f g (closure s)

If two continuous maps are equal on s, then they are equal on the closure of s.

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theorem continuous.ext_on {α : Type u} {β : Type v} [topological_space α] [topological_space β] [t2_space α] {s : set β} (hs : dense s) {f g : β → α} (hf : continuous f) (hg : continuous g) (h : set.eq_on f g s) :

f = g

If two continuous functions are equal on a dense set, then they are equal.

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theorem function.left_inverse.closed_range {α : Type u} {β : Type v} [topological_space α] [topological_space β] [t2_space α] {f : α → β} {g : β → α} (h : function.left_inverse f g) (hf : continuous f) (hg : continuous g) :

is_closed (set.range g)

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theorem function.left_inverse.closed_embedding {α : Type u} {β : Type v} [topological_space α] [topological_space β] [t2_space α] {f : α → β} {g : β → α} (h : function.left_inverse f g) (hf : continuous f) (hg : continuous g) :

closed_embedding g

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theorem diagonal_eq_range_diagonal_map {α : Type u_1} :

{p : α × α | p.fst = p.snd} = set.range (λ (x : α), (x, x))

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theorem prod_subset_compl_diagonal_iff_disjoint {α : Type u_1} {s t : set α} :

s ×ˢ t ⊆ {p : α × α | p.fst = p.snd}ᶜ ↔ s ∩ t = ∅

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theorem compact_compact_separated {α : Type u} [topological_space α] [t2_space α] {s t : set α} (hs : is_compact s) (ht : is_compact t) (hst : s ∩ t = ∅) :

∃ (u v : set α), is_open u ∧ is_open v ∧ s ⊆ u ∧ t ⊆ v ∧ u ∩ v = ∅

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theorem is_compact.is_closed {α : Type u} [topological_space α] [t2_space α] {s : set α} (hs : is_compact s) :

is_closed s

In a t2_space, every compact set is closed.

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@[simp]

theorem filter.coclosed_compact_eq_cocompact {α : Type u} [topological_space α] [t2_space α] :

filter.coclosed_compact α = filter.cocompact α

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theorem exists_subset_nhd_of_compact {α : Type u} [topological_space α] [t2_space α] {ι : Type u_1} [nonempty ι] {V : ι → set α} (hV : directed superset V) (hV_cpct : ∀ (i : ι), is_compact (V i)) {U : set α} (hU : ∀ (x : α), (x ∈ ⋂ (i : ι), V i) → U ∈ 𝓝 x) :

∃ (i : ι), V i ⊆ U

If V : ι → set α is a decreasing family of compact sets then any neighborhood of ⋂ i, V i contains some V i. This is a version of exists_subset_nhd_of_compact' where we don't need to assume each V i closed because it follows from compactness since α is assumed to be Hausdorff.

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theorem compact_exhaustion.is_closed {α : Type u} [topological_space α] [t2_space α] (K : compact_exhaustion α) (n : ℕ) :

is_closed (⇑K n)

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theorem is_compact.inter {α : Type u} [topological_space α] [t2_space α] {s t : set α} (hs : is_compact s) (ht : is_compact t) :

is_compact (s ∩ t)

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theorem compact_closure_of_subset_compact {α : Type u} [topological_space α] [t2_space α] {s t : set α} (ht : is_compact t) (h : s ⊆ t) :

is_compact (closure s)

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theorem image_closure_of_compact {α : Type u} {β : Type v} [topological_space α] [topological_space β] [t2_space β] {s : set α} (hs : is_compact (closure s)) {f : α → β} (hf : continuous_on f (closure s)) :

f '' closure s = closure (f '' s)

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theorem is_compact.binary_compact_cover {α : Type u} [topological_space α] [t2_space α] {K U V : set α} (hK : is_compact K) (hU : is_open U) (hV : is_open V) (h2K : K ⊆ U ∪ V) :

∃ (K₁ K₂ : set α), is_compact K₁ ∧ is_compact K₂ ∧ K₁ ⊆ U ∧ K₂ ⊆ V ∧ K = K₁ ∪ K₂

If a compact set is covered by two open sets, then we can cover it by two compact subsets.

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theorem continuous.is_closed_map {α : Type u} {β : Type v} [topological_space α] [topological_space β] [compact_space α] [t2_space β] {f : α → β} (h : continuous f) :

is_closed_map f

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theorem continuous.closed_embedding {α : Type u} {β : Type v} [topological_space α] [topological_space β] [compact_space α] [t2_space β] {f : α → β} (h : continuous f) (hf : function.injective f) :

closed_embedding f

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theorem is_compact.finite_compact_cover {α : Type u} [topological_space α] [t2_space α] {s : set α} (hs : is_compact s) {ι : Type u_1} (t : finset ι) (U : ι → set α) (hU : ∀ (i : ι), i ∈ t → is_open (U i)) (hsC : s ⊆ ⋃ (i : ι) (H : i ∈ t), U i) :

∃ (K : ι → set α), (∀ (i : ι), is_compact (K i)) ∧ (∀ (i : ι), K i ⊆ U i) ∧ s = ⋃ (i : ι) (H : i ∈ t), K i

For every finite open cover Uᵢ of a compact set, there exists a compact cover Kᵢ ⊆ Uᵢ.

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theorem locally_compact_of_compact_nhds {α : Type u} [topological_space α] [t2_space α] (h : ∀ (x : α), ∃ (s : set α), s ∈ 𝓝 x ∧ is_compact s) :

locally_compact_space α

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@[protected, instance]

def locally_compact_of_compact {α : Type u} [topological_space α] [t2_space α] [compact_space α] :

locally_compact_space α

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theorem exists_open_with_compact_closure {α : Type u} [topological_space α] [locally_compact_space α] [t2_space α] (x : α) :

∃ (U : set α), is_open U ∧ x ∈ U ∧ is_compact (closure U)

In a locally compact T₂ space, every point has an open neighborhood with compact closure

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theorem exists_open_superset_and_is_compact_closure {α : Type u} [topological_space α] [locally_compact_space α] [t2_space α] {K : set α} (hK : is_compact K) :

∃ (V : set α), is_open V ∧ K ⊆ V ∧ is_compact (closure V)

In a locally compact T₂ space, every compact set has an open neighborhood with compact closure.

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theorem is_preirreducible_iff_subsingleton {α : Type u} [topological_space α] [t2_space α] (S : set α) :

is_preirreducible S ↔ subsingleton ↥S

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theorem is_irreducible_iff_singleton {α : Type u} [topological_space α] [t2_space α] (S : set α) :

is_irreducible S ↔ ∃ (x : α), S = {x}

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@[class]

structure regular_space (α : Type u) [topological_space α] :

Prop

to_t0_space : t0_space α
regular : ∀ {s : set α} {a : α}, is_closed s → a ∉ s → (∃ (t : set α), is_open t ∧ s ⊆ t ∧ 𝓝[t] a = ⊥)

A T₃ space, also known as a regular space (although this condition sometimes omits T₂), is one in which for every closed C and x ∉ C, there exist disjoint open sets containing x and C respectively.

Instances

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@[protected, instance]

def regular_space.t1_space {α : Type u} [topological_space α] [regular_space α] :

t1_space α

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theorem nhds_is_closed {α : Type u} [topological_space α] [regular_space α] {a : α} {s : set α} (h : s ∈ 𝓝 a) :

∃ (t : set α) (H : t ∈ 𝓝 a), t ⊆ s ∧ is_closed t

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theorem closed_nhds_basis {α : Type u} [topological_space α] [regular_space α] (a : α) :

(𝓝 a).has_basis (λ (s : set α), s ∈ 𝓝 a ∧ is_closed s) id

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theorem topological_space.is_topological_basis.exists_closure_subset {α : Type u} [topological_space α] [regular_space α] {B : set (set α)} (hB : topological_space.is_topological_basis B) {a : α} {s : set α} (h : s ∈ 𝓝 a) :

∃ (t : set α) (H : t ∈ B), a ∈ t ∧ closure t ⊆ s

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theorem topological_space.is_topological_basis.nhds_basis_closure {α : Type u} [topological_space α] [regular_space α] {B : set (set α)} (hB : topological_space.is_topological_basis B) (a : α) :

(𝓝 a).has_basis (λ (s : set α), a ∈ s ∧ s ∈ B) closure

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@[protected, instance]

def subtype.regular_space {α : Type u} [topological_space α] [regular_space α] {p : α → Prop} :

regular_space (subtype p)

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@[protected, instance]

def regular_space.t2_space (α : Type u) [topological_space α] [regular_space α] :

t2_space α

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@[protected, instance]

def regular_space.t2_5_space (α : Type u) [topological_space α] [regular_space α] :

t2_5_space α

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theorem disjoint_nested_nhds {α : Type u} [topological_space α] [regular_space α] {x y : α} (h : x ≠ y) :

∃ (U₁ : set α) (H : U₁ ∈ 𝓝 x) (V₁ : set α) (H : V₁ ∈ 𝓝 x) (U₂ : set α) (H : U₂ ∈ 𝓝 y) (V₂ : set α) (H : V₂ ∈ 𝓝 y), is_closed V₁ ∧ is_closed V₂ ∧ is_open U₁ ∧ is_open U₂ ∧ V₁ ⊆ U₁ ∧ V₂ ⊆ U₂ ∧ U₁ ∩ U₂ = ∅

Given two points x ≠ y, we can find neighbourhoods x ∈ V₁ ⊆ U₁ and y ∈ V₂ ⊆ U₂, with the Vₖ closed and the Uₖ open, such that the Uₖ are disjoint.

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theorem exists_compact_between {α : Type u} [topological_space α] [locally_compact_space α] [regular_space α] {K U : set α} (hK : is_compact K) (hU : is_open U) (hKU : K ⊆ U) :

∃ (K' : set α), is_compact K' ∧ K ⊆ interior K' ∧ K' ⊆ U

In a locally compact regular space, given a compact set K inside an open set U, we can find a compact set K' between these sets: K is inside the interior of K' and K' ⊆ U.

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theorem exists_open_between_and_is_compact_closure {α : Type u} [topological_space α] [locally_compact_space α] [regular_space α] {K U : set α} (hK : is_compact K) (hU : is_open U) (hKU : K ⊆ U) :

∃ (V : set α), is_open V ∧ K ⊆ V ∧ closure V ⊆ U ∧ is_compact (closure V)

In a locally compact regular space, given a compact set K inside an open set U, we can find a open set V between these sets with compact closure: K ⊆ V and the closure of V is inside U.

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@[class]

structure normal_space (α : Type u) [topological_space α] :

Prop

to_t1_space : t1_space α
normal : ∀ (s t : set α), is_closed s → is_closed t → disjoint s t → (∃ (u v : set α), is_open u ∧ is_open v ∧ s ⊆ u ∧ t ⊆ v ∧ disjoint u v)

A T₄ space, also known as a normal space (although this condition sometimes omits T₂), is one in which for every pair of disjoint closed sets C and D, there exist disjoint open sets containing C and D respectively.

Instances

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theorem normal_separation {α : Type u} [topological_space α] [normal_space α] {s t : set α} (H1 : is_closed s) (H2 : is_closed t) (H3 : disjoint s t) :

∃ (u v : set α), is_open u ∧ is_open v ∧ s ⊆ u ∧ t ⊆ v ∧ disjoint u v

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theorem normal_exists_closure_subset {α : Type u} [topological_space α] [normal_space α] {s t : set α} (hs : is_closed s) (ht : is_open t) (hst : s ⊆ t) :

∃ (u : set α), is_open u ∧ s ⊆ u ∧ closure u ⊆ t

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@[protected, instance]

def normal_space.regular_space {α : Type u} [topological_space α] [normal_space α] :

regular_space α

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theorem normal_of_compact_t2 {α : Type u} [topological_space α] [compact_space α] [t2_space α] :

normal_space α

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theorem normal_space_of_regular_second_countable (α : Type u) [topological_space α] [topological_space.second_countable_topology α] [regular_space α] :

normal_space α

A regular topological space with second countable topology is a normal space. This lemma is not an instance to avoid a loop.

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theorem connected_component_eq_Inter_clopen {α : Type u} [topological_space α] [t2_space α] [compact_space α] (x : α) :

connected_component x = ⋂ (Z : {Z // is_clopen Z ∧ x ∈ Z}), ↑Z

In a compact t2 space, the connected component of a point equals the intersection of all its clopen neighbourhoods.

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theorem tot_sep_of_zero_dim {α : Type u} [topological_space α] [t2_space α] (h : topological_space.is_topological_basis {s : set α | is_clopen s}) :

totally_separated_space α

A Hausdorff space with a clopen basis is totally separated.

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theorem compact_t2_tot_disc_iff_tot_sep {α : Type u} [topological_space α] [t2_space α] [compact_space α] :

totally_disconnected_space α ↔ totally_separated_space α

A compact Hausdorff space is totally disconnected if and only if it is totally separated, this is also true for locally compact spaces.

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theorem nhds_basis_clopen {α : Type u} [topological_space α] [t2_space α] [compact_space α] [totally_disconnected_space α] (x : α) :

(𝓝 x).has_basis (λ (s : set α), x ∈ s ∧ is_clopen s) id

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theorem is_topological_basis_clopen {α : Type u} [topological_space α] [t2_space α] [compact_space α] [totally_disconnected_space α] :

topological_space.is_topological_basis {s : set α | is_clopen s}

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theorem compact_exists_clopen_in_open {α : Type u} [topological_space α] [t2_space α] [compact_space α] [totally_disconnected_space α] {x : α} {U : set α} (is_open : _root_.is_open U) (memU : x ∈ U) :

∃ (V : set α) (hV : is_clopen V), x ∈ V ∧ V ⊆ U

Every member of an open set in a compact Hausdorff totally disconnected space is contained in a clopen set contained in the open set.

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theorem loc_compact_Haus_tot_disc_of_zero_dim {H : Type u_1} [topological_space H] [locally_compact_space H] [t2_space H] [totally_disconnected_space H] :

topological_space.is_topological_basis {s : set H | is_clopen s}

A locally compact Hausdorff totally disconnected space has a basis with clopen elements.

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theorem loc_compact_t2_tot_disc_iff_tot_sep {H : Type u_1} [topological_space H] [locally_compact_space H] [t2_space H] :

totally_disconnected_space H ↔ totally_separated_space H

A locally compact Hausdorff space is totally disconnected if and only if it is totally separated.

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@[protected, instance]

def connected_components.t2 {α : Type u} [topological_space α] [t2_space α] [compact_space α] :

t2_space (connected_components α)

connected_components α is Hausdorff when α is Hausdorff and compact

Separation properties of topological spaces. #

Main definitions #

Main results #

T₀ spaces #

T₁ spaces #

T₂ spaces #

T₃ spaces #

Discrete spaces #

References #

Properties of Lim and lim #

t2_space constructions #

Properties of `Lim` and `lim` #

`t2_space` constructions #