statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
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insert_mem_nhds_within_of_subset_insert [t1_space α] {x y : α} {s t : set α}
(hu : t ⊆ insert y s) :
insert x s ∈ 𝓝[t] x | begin
rcases eq_or_ne x y with rfl|h,
{ exact mem_of_superset self_mem_nhds_within hu },
refine nhds_within_mono x hu _,
rw [nhds_within_insert_of_ne h],
exact mem_of_superset self_mem_nhds_within (subset_insert x s)
end | lemma | insert_mem_nhds_within_of_subset_insert | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"eq_or_ne",
"nhds_within_insert_of_ne",
"nhds_within_mono",
"self_mem_nhds_within",
"t1_space"
] | If `t` is a subset of `s`, except for one point,
then `insert x s` is a neighborhood of `x` within `t`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bInter_basis_nhds [t1_space α] {ι : Sort*} {p : ι → Prop} {s : ι → set α} {x : α}
(h : (𝓝 x).has_basis p s) : (⋂ i (h : p i), s i) = {x} | begin
simp only [eq_singleton_iff_unique_mem, mem_Inter],
refine ⟨λ i hi, mem_of_mem_nhds $ h.mem_of_mem hi, λ y hy, _⟩,
contrapose! hy,
rcases h.mem_iff.1 (compl_singleton_mem_nhds hy.symm) with ⟨i, hi, hsub⟩,
exact ⟨i, hi, λ h, hsub h rfl⟩
end | lemma | bInter_basis_nhds | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"compl_singleton_mem_nhds",
"mem_of_mem_nhds",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_singleton_mem_nhds_set_iff [t1_space α] {x : α} {s : set α} :
{x}ᶜ ∈ 𝓝ˢ s ↔ x ∉ s | by rwa [is_open_compl_singleton.mem_nhds_set, subset_compl_singleton_iff] | lemma | compl_singleton_mem_nhds_set_iff | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_set_le_iff [t1_space α] {s t : set α} : 𝓝ˢ s ≤ 𝓝ˢ t ↔ s ⊆ t | begin
refine ⟨_, λ h, monotone_nhds_set h⟩,
simp_rw [filter.le_def], intros h x hx,
specialize h {x}ᶜ,
simp_rw [compl_singleton_mem_nhds_set_iff] at h,
by_contra hxt,
exact h hxt hx,
end | lemma | nhds_set_le_iff | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"by_contra",
"compl_singleton_mem_nhds_set_iff",
"filter.le_def",
"monotone_nhds_set",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_set_inj_iff [t1_space α] {s t : set α} : 𝓝ˢ s = 𝓝ˢ t ↔ s = t | by { simp_rw [le_antisymm_iff], exact and_congr nhds_set_le_iff nhds_set_le_iff } | lemma | nhds_set_inj_iff | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"nhds_set_le_iff",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective_nhds_set [t1_space α] : function.injective (𝓝ˢ : set α → filter α) | λ s t hst, nhds_set_inj_iff.mp hst | lemma | injective_nhds_set | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"filter",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_nhds_set [t1_space α] : strict_mono (𝓝ˢ : set α → filter α) | monotone_nhds_set.strict_mono_of_injective injective_nhds_set | lemma | strict_mono_nhds_set | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"filter",
"injective_nhds_set",
"strict_mono",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_le_nhds_set_iff [t1_space α] {s : set α} {x : α} : 𝓝 x ≤ 𝓝ˢ s ↔ x ∈ s | by rw [← nhds_set_singleton, nhds_set_le_iff, singleton_subset_iff] | lemma | nhds_le_nhds_set_iff | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"nhds_set_le_iff",
"nhds_set_singleton",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense.diff_singleton [t1_space α] {s : set α} (hs : dense s) (x : α) [ne_bot (𝓝[≠] x)] :
dense (s \ {x}) | hs.inter_of_open_right (dense_compl_singleton x) is_open_compl_singleton | lemma | dense.diff_singleton | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"dense",
"dense_compl_singleton",
"is_open_compl_singleton",
"t1_space"
] | Removing a non-isolated point from a dense set, one still obtains a dense set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense.diff_finset [t1_space α] [∀ (x : α), ne_bot (𝓝[≠] x)]
{s : set α} (hs : dense s) (t : finset α) :
dense (s \ t) | begin
induction t using finset.induction_on with x s hxs ih hd,
{ simpa using hs },
{ rw [finset.coe_insert, ← union_singleton, ← diff_diff],
exact ih.diff_singleton _, }
end | lemma | dense.diff_finset | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"dense",
"finset",
"finset.coe_insert",
"finset.induction_on",
"ih",
"t1_space"
] | Removing a finset from a dense set in a space without isolated points, one still
obtains a dense set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense.diff_finite [t1_space α] [∀ (x : α), ne_bot (𝓝[≠] x)]
{s : set α} (hs : dense s) {t : set α} (ht : t.finite) :
dense (s \ t) | begin
convert hs.diff_finset ht.to_finset,
exact (finite.coe_to_finset _).symm,
end | lemma | dense.diff_finite | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"dense",
"t1_space"
] | Removing a finite set from a dense set in a space without isolated points, one still
obtains a dense set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_of_tendsto_nhds [topological_space β] [t1_space β] {f : α → β} {a : α} {b : β}
(h : tendsto f (𝓝 a) (𝓝 b)) : f a = b | by_contra $ assume (hfa : f a ≠ b),
have fact₁ : {f a}ᶜ ∈ 𝓝 b := compl_singleton_mem_nhds hfa.symm,
have fact₂ : tendsto f (pure a) (𝓝 b) := h.comp (tendsto_id'.2 $ pure_le_nhds a),
fact₂ fact₁ (eq.refl $ f a) | lemma | eq_of_tendsto_nhds | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"by_contra",
"compl_singleton_mem_nhds",
"pure_le_nhds",
"t1_space",
"topological_space"
] | If a function to a `t1_space` tends to some limit `b` at some point `a`, then necessarily
`b = f a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.tendsto.eventually_ne [topological_space β] [t1_space β] {α : Type*} {g : α → β}
{l : filter α} {b₁ b₂ : β} (hg : tendsto g l (𝓝 b₁)) (hb : b₁ ≠ b₂) :
∀ᶠ z in l, g z ≠ b₂ | hg.eventually (is_open_compl_singleton.eventually_mem hb) | lemma | filter.tendsto.eventually_ne | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"filter",
"t1_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.eventually_ne [topological_space β] [t1_space β] {g : α → β}
{a : α} {b : β} (hg1 : continuous_at g a) (hg2 : g a ≠ b) :
∀ᶠ z in 𝓝 a, g z ≠ b | hg1.tendsto.eventually_ne hg2 | lemma | continuous_at.eventually_ne | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"continuous_at",
"t1_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_of_tendsto_nhds [topological_space β] [t1_space β] {f : α → β} {a : α} {b : β}
(h : tendsto f (𝓝 a) (𝓝 b)) : continuous_at f a | show tendsto f (𝓝 a) (𝓝 $ f a), by rwa eq_of_tendsto_nhds h | lemma | continuous_at_of_tendsto_nhds | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"continuous_at",
"eq_of_tendsto_nhds",
"t1_space",
"topological_space"
] | 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`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_const_nhds_iff [t1_space α] {l : filter β} [ne_bot l] {c d : α} :
tendsto (λ x, c) l (𝓝 d) ↔ c = d | by simp_rw [tendsto, filter.map_const, pure_le_nhds_iff] | lemma | tendsto_const_nhds_iff | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"filter",
"filter.map_const",
"pure_le_nhds_iff",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_singleton_of_finite_mem_nhds {α : Type*} [topological_space α] [t1_space α]
(x : α) {s : set α} (hs : s ∈ 𝓝 x) (hsf : s.finite) : is_open ({x} : set α) | begin
have A : {x} ⊆ s, by simp only [singleton_subset_iff, mem_of_mem_nhds hs],
have B : is_closed (s \ {x}) := (hsf.subset (diff_subset _ _)).is_closed,
have C : (s \ {x})ᶜ ∈ 𝓝 x, from B.is_open_compl.mem_nhds (λ h, h.2 rfl),
have D : {x} ∈ 𝓝 x, by simpa only [← diff_eq, diff_diff_cancel_left A] using inter... | lemma | is_open_singleton_of_finite_mem_nhds | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"is_closed",
"is_open",
"mem_interior_iff_mem_nhds",
"mem_of_mem_nhds",
"subset_interior_iff_is_open",
"t1_space",
"topological_space"
] | A point with a finite neighborhood has to be isolated. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
infinite_of_mem_nhds {α} [topological_space α] [t1_space α] (x : α) [hx : ne_bot (𝓝[≠] x)]
{s : set α} (hs : s ∈ 𝓝 x) : set.infinite s | begin
refine λ hsf, hx.1 _,
rw [← is_open_singleton_iff_punctured_nhds],
exact is_open_singleton_of_finite_mem_nhds x hs hsf
end | lemma | infinite_of_mem_nhds | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"is_open_singleton_iff_punctured_nhds",
"is_open_singleton_of_finite_mem_nhds",
"set.infinite",
"t1_space",
"topological_space"
] | If the punctured neighborhoods of a point form a nontrivial filter, then any neighborhood is
infinite. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
discrete_of_t1_of_finite {X : Type*} [topological_space X] [t1_space X] [finite X] :
discrete_topology X | begin
apply singletons_open_iff_discrete.mp,
intros x,
rw [← is_closed_compl_iff],
exact (set.to_finite _).is_closed
end | lemma | discrete_of_t1_of_finite | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"discrete_topology",
"finite",
"is_closed",
"is_closed_compl_iff",
"set.to_finite",
"t1_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preconnected_space.trivial_of_discrete [preconnected_space α] [discrete_topology α] :
subsingleton α | begin
rw ←not_nontrivial_iff_subsingleton,
rintro ⟨x, y, hxy⟩,
rw [ne.def, ←mem_singleton_iff, (is_clopen_discrete _).eq_univ $ singleton_nonempty y] at hxy,
exact hxy (mem_univ x)
end | lemma | preconnected_space.trivial_of_discrete | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"discrete_topology",
"is_clopen_discrete",
"preconnected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected.infinite_of_nontrivial [t1_space α] {s : set α} (h : is_preconnected s)
(hs : s.nontrivial) : s.infinite | begin
refine mt (λ hf, (subsingleton_coe s).mp _) (not_subsingleton_iff.mpr hs),
haveI := @discrete_of_t1_of_finite s _ _ hf.to_subtype,
exact @preconnected_space.trivial_of_discrete _ _ (subtype.preconnected_space h) _
end | lemma | is_preconnected.infinite_of_nontrivial | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"discrete_of_t1_of_finite",
"is_preconnected",
"preconnected_space.trivial_of_discrete",
"subtype.preconnected_space",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
connected_space.infinite [connected_space α] [nontrivial α] [t1_space α] : infinite α | infinite_univ_iff.mp $ is_preconnected_univ.infinite_of_nontrivial nontrivial_univ | lemma | connected_space.infinite | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"connected_space",
"infinite",
"nontrivial",
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
singleton_mem_nhds_within_of_mem_discrete {s : set α} [discrete_topology s]
{x : α} (hx : x ∈ s) :
{x} ∈ 𝓝[s] x | begin
have : ({⟨x, hx⟩} : set s) ∈ 𝓝 (⟨x, hx⟩ : s), by simp [nhds_discrete],
simpa only [nhds_within_eq_map_subtype_coe hx, image_singleton]
using @image_mem_map _ _ _ (coe : s → α) _ this
end | lemma | singleton_mem_nhds_within_of_mem_discrete | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"discrete_topology",
"nhds_discrete",
"nhds_within_eq_map_subtype_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_of_mem_discrete {s : set α} [discrete_topology s] {x : α} (hx : x ∈ s) :
𝓝[s] x = pure x | le_antisymm (le_pure_iff.2 $ singleton_mem_nhds_within_of_mem_discrete hx) (pure_le_nhds_within hx) | lemma | nhds_within_of_mem_discrete | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"discrete_topology",
"pure_le_nhds_within",
"singleton_mem_nhds_within_of_mem_discrete"
] | 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}`.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.has_basis.exists_inter_eq_singleton_of_mem_discrete
{ι : Type*} {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 (hi : p i), t i ∩ s = {x} | begin
rcases (nhds_within_has_basis hb s).mem_iff.1 (singleton_mem_nhds_within_of_mem_discrete hx)
with ⟨i, hi, hix⟩,
exact ⟨i, hi, subset.antisymm hix $ singleton_subset_iff.2
⟨mem_of_mem_nhds $ hb.mem_of_mem hi, hx⟩⟩
end | lemma | filter.has_basis.exists_inter_eq_singleton_of_mem_discrete | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"discrete_topology",
"nhds_within_has_basis",
"singleton_mem_nhds_within_of_mem_discrete"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_inter_eq_singleton_of_mem_discrete {s : set α} [discrete_topology s]
{x : α} (hx : x ∈ s) :
∃ U ∈ 𝓝 x, U ∩ s = {x} | by simpa using (𝓝 x).basis_sets.exists_inter_eq_singleton_of_mem_discrete hx | lemma | nhds_inter_eq_singleton_of_mem_discrete | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"discrete_topology"
] | A point `x` in a discrete subset `s` of a topological space admits a neighbourhood
that only meets `s` at `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
disjoint_nhds_within_of_mem_discrete {s : set α} [discrete_topology s] {x : α} (hx : x ∈ s) :
∃ U ∈ 𝓝[≠] x, disjoint U s | let ⟨V, h, h'⟩ := nhds_inter_eq_singleton_of_mem_discrete hx in
⟨{x}ᶜ ∩ V, inter_mem_nhds_within _ h,
(disjoint_iff_inter_eq_empty.mpr (by { rw [inter_assoc, h', compl_inter_self] }))⟩ | lemma | disjoint_nhds_within_of_mem_discrete | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"discrete_topology",
"disjoint",
"inter_mem_nhds_within",
"nhds_inter_eq_singleton_of_mem_discrete"
] | For point `x` in a discrete subset `s` of a topological space, there is a set `U`
such that
1. `U` is a punctured neighborhood of `x` (ie. `U ∪ {x}` is a neighbourhood of `x`),
2. `U` is disjoint from `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
topological_space.subset_trans {X : Type*} [tX : topological_space X]
{s t : set X} (ts : t ⊆ s) :
(subtype.topological_space : topological_space t) =
(subtype.topological_space : topological_space s).induced (set.inclusion ts) | begin
change tX.induced ((coe : s → X) ∘ (set.inclusion ts)) =
topological_space.induced (set.inclusion ts) (tX.induced _),
rw ← induced_compose,
end | lemma | topological_space.subset_trans | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"induced_compose",
"set.inclusion",
"topological_space",
"topological_space.induced"
] | 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`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
t2_space (α : Type u) [topological_space α] : Prop | (t2 : ∀ x y, x ≠ y → ∃ u v : set α, is_open u ∧ is_open v ∧ x ∈ u ∧ y ∈ v ∧ disjoint u v) | class | t2_space | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"disjoint",
"is_open",
"topological_space"
] | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
t2_separation [t2_space α] {x y : α} (h : x ≠ y) :
∃ u v : set α, is_open u ∧ is_open v ∧ x ∈ u ∧ y ∈ v ∧ disjoint u v | t2_space.t2 x y h | lemma | t2_separation | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"disjoint",
"is_open",
"t2_space"
] | Two different points can be separated by open sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
t2_space_iff_disjoint_nhds : t2_space α ↔ ∀ x y : α, x ≠ y → disjoint (𝓝 x) (𝓝 y) | begin
refine (t2_space_iff α).trans (forall₃_congr $ λ x y hne, _),
simp only [(nhds_basis_opens x).disjoint_iff (nhds_basis_opens y), exists_prop,
← exists_and_distrib_left, and.assoc, and_comm, and.left_comm]
end | lemma | t2_space_iff_disjoint_nhds | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"disjoint",
"disjoint_iff",
"exists_and_distrib_left",
"exists_prop",
"forall₃_congr",
"nhds_basis_opens",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_nhds_nhds [t2_space α] {x y : α} : disjoint (𝓝 x) (𝓝 y) ↔ x ≠ y | ⟨λ hd he, by simpa [he, nhds_ne_bot.ne] using hd, t2_space_iff_disjoint_nhds.mp ‹_› x y⟩ | lemma | disjoint_nhds_nhds | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"disjoint",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pairwise_disjoint_nhds [t2_space α] : pairwise (disjoint on (𝓝 : α → filter α)) | λ x y, disjoint_nhds_nhds.2 | lemma | pairwise_disjoint_nhds | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"disjoint",
"filter",
"pairwise",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.pairwise_disjoint_nhds [t2_space α] (s : set α) : s.pairwise_disjoint 𝓝 | pairwise_disjoint_nhds.set_pairwise s | lemma | set.pairwise_disjoint_nhds | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.finite.t2_separation [t2_space α] {s : set α} (hs : s.finite) :
∃ U : α → set α, (∀ x, x ∈ U x ∧ is_open (U x)) ∧ s.pairwise_disjoint U | s.pairwise_disjoint_nhds.exists_mem_filter_basis hs nhds_basis_opens | lemma | set.finite.t2_separation | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"is_open",
"nhds_basis_opens",
"t2_space"
] | Points of a finite set can be separated by open sets from each other. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_set_of_disjoint_nhds_nhds :
is_open {p : α × α | disjoint (𝓝 p.1) (𝓝 p.2)} | begin
simp only [is_open_iff_mem_nhds, prod.forall, mem_set_of_eq],
intros x y h,
obtain ⟨U, hU, V, hV, hd⟩ := ((nhds_basis_opens x).disjoint_iff (nhds_basis_opens y)).mp h,
exact mem_nhds_prod_iff.mpr ⟨U, hU.2.mem_nhds hU.1, V, hV.2.mem_nhds hV.1,
λ ⟨x', y'⟩ ⟨hx', hy'⟩, disjoint_of_disjoint_of_mem hd (hU.2... | lemma | is_open_set_of_disjoint_nhds_nhds | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"disjoint",
"disjoint_iff",
"is_open",
"is_open_iff_mem_nhds",
"nhds_basis_opens"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t2_space.t1_space [t2_space α] : t1_space α | t1_space_iff_disjoint_pure_nhds.mpr $ λ x y hne, (disjoint_nhds_nhds.2 hne).mono_left $
pure_le_nhds _ | instance | t2_space.t1_space | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"pure_le_nhds",
"t1_space",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t2_iff_nhds : t2_space α ↔ ∀ {x y : α}, ne_bot (𝓝 x ⊓ 𝓝 y) → x = y | by simp only [t2_space_iff_disjoint_nhds, disjoint_iff, ne_bot_iff, ne.def, not_imp_comm] | lemma | t2_iff_nhds | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"disjoint_iff",
"not_imp_comm",
"t2_space",
"t2_space_iff_disjoint_nhds"
] | A space is T₂ iff the neighbourhoods of distinct points generate the bottom filter. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_of_nhds_ne_bot [t2_space α] {x y : α} (h : ne_bot (𝓝 x ⊓ 𝓝 y)) : x = y | t2_iff_nhds.mp ‹_› h | lemma | eq_of_nhds_ne_bot | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t2_space_iff_nhds : t2_space α ↔ ∀ {x y : α}, x ≠ y → ∃ (U ∈ 𝓝 x) (V ∈ 𝓝 y), disjoint U V | by simp only [t2_space_iff_disjoint_nhds, filter.disjoint_iff] | lemma | t2_space_iff_nhds | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"disjoint",
"filter.disjoint_iff",
"t2_space",
"t2_space_iff_disjoint_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t2_separation_nhds [t2_space α] {x y : α} (h : x ≠ y) :
∃ u v, u ∈ 𝓝 x ∧ v ∈ 𝓝 y ∧ disjoint u v | let ⟨u, v, open_u, open_v, x_in, y_in, huv⟩ := t2_separation h in
⟨u, v, open_u.mem_nhds x_in, open_v.mem_nhds y_in, huv⟩ | lemma | t2_separation_nhds | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"disjoint",
"t2_separation",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t2_separation_compact_nhds [locally_compact_space α] [t2_space α] {x y : α} (h : x ≠ y) :
∃ u v, u ∈ 𝓝 x ∧ v ∈ 𝓝 y ∧ is_compact u ∧ is_compact v ∧ disjoint u v | by simpa only [exists_prop, ← exists_and_distrib_left, and_comm, and.assoc, and.left_comm]
using ((compact_basis_nhds x).disjoint_iff (compact_basis_nhds y)).1 (disjoint_nhds_nhds.2 h) | lemma | t2_separation_compact_nhds | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"compact_basis_nhds",
"disjoint",
"disjoint_iff",
"exists_and_distrib_left",
"exists_prop",
"is_compact",
"locally_compact_space",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t2_iff_ultrafilter :
t2_space α ↔ ∀ {x y : α} (f : ultrafilter α), ↑f ≤ 𝓝 x → ↑f ≤ 𝓝 y → x = y | t2_iff_nhds.trans $ by simp only [←exists_ultrafilter_iff, and_imp, le_inf_iff, exists_imp_distrib] | lemma | t2_iff_ultrafilter | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"and_imp",
"exists_imp_distrib",
"le_inf_iff",
"t2_space",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t2_iff_is_closed_diagonal : t2_space α ↔ is_closed (diagonal α) | by simp only [t2_space_iff_disjoint_nhds, ← is_open_compl_iff, is_open_iff_mem_nhds, prod.forall,
nhds_prod_eq, compl_diagonal_mem_prod, mem_compl_iff, mem_diagonal_iff] | lemma | t2_iff_is_closed_diagonal | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"is_closed",
"is_open_compl_iff",
"is_open_iff_mem_nhds",
"nhds_prod_eq",
"t2_space",
"t2_space_iff_disjoint_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_diagonal [t2_space α] : is_closed (diagonal α) | t2_iff_is_closed_diagonal.mp ‹_› | lemma | is_closed_diagonal | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"is_closed",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset_disjoint_finset_opens_of_t2 [t2_space α] :
∀ (s t : finset α), disjoint s t → separated_nhds (s : set α) t | begin
refine induction_on_union _ (λ a b hi d, (hi d.symm).symm) (λ a d, empty_right a) (λ a b ab, _) _,
{ obtain ⟨U, V, oU, oV, aU, bV, UV⟩ := t2_separation (finset.disjoint_singleton.1 ab),
refine ⟨U, V, oU, oV, _, _, UV⟩;
exact singleton_subset_set_iff.mpr ‹_› },
{ intros a b c ac bc d,
apply_mod_c... | lemma | finset_disjoint_finset_opens_of_t2 | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"disjoint",
"finset",
"separated_nhds",
"t2_separation",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
point_disjoint_finset_opens_of_t2 [t2_space α] {x : α} {s : finset α} (h : x ∉ s) :
separated_nhds ({x} : set α) s | by exact_mod_cast finset_disjoint_finset_opens_of_t2 {x} s (finset.disjoint_singleton_left.mpr h) | lemma | point_disjoint_finset_opens_of_t2 | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"finset",
"finset_disjoint_finset_opens_of_t2",
"separated_nhds",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_unique [t2_space α] {f : β → α} {l : filter β} {a b : α}
[ne_bot l] (ha : tendsto f l (𝓝 a)) (hb : tendsto f l (𝓝 b)) : a = b | eq_of_nhds_ne_bot $ ne_bot_of_le $ le_inf ha hb | lemma | tendsto_nhds_unique | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"eq_of_nhds_ne_bot",
"filter",
"le_inf",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_unique' [t2_space α] {f : β → α} {l : filter β} {a b : α}
(hl : ne_bot l) (ha : tendsto f l (𝓝 a)) (hb : tendsto f l (𝓝 b)) : a = b | eq_of_nhds_ne_bot $ ne_bot_of_le $ le_inf ha hb | lemma | tendsto_nhds_unique' | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"eq_of_nhds_ne_bot",
"filter",
"le_inf",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_unique_of_eventually_eq [t2_space α] {f g : β → α} {l : filter β} {a b : α}
[ne_bot l] (ha : tendsto f l (𝓝 a)) (hb : tendsto g l (𝓝 b)) (hfg : f =ᶠ[l] g) :
a = b | tendsto_nhds_unique (ha.congr' hfg) hb | lemma | tendsto_nhds_unique_of_eventually_eq | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"filter",
"t2_space",
"tendsto_nhds_unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_unique_of_frequently_eq [t2_space α] {f g : β → α} {l : filter β} {a b : α}
(ha : tendsto f l (𝓝 a)) (hb : tendsto g l (𝓝 b)) (hfg : ∃ᶠ x in l, f x = g x) :
a = b | have ∃ᶠ z : α × α in 𝓝 (a, b), z.1 = z.2 := (ha.prod_mk_nhds hb).frequently hfg,
not_not.1 $ λ hne, this (is_closed_diagonal.is_open_compl.mem_nhds hne) | lemma | tendsto_nhds_unique_of_frequently_eq | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"filter",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t2_5_space (α : Type u) [topological_space α]: Prop | (t2_5 : ∀ ⦃x y : α⦄ (h : x ≠ y), disjoint ((𝓝 x).lift' closure) ((𝓝 y).lift' closure)) | class | t2_5_space | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closure",
"disjoint",
"topological_space"
] | 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` . | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
disjoint_lift'_closure_nhds [t2_5_space α] {x y : α} :
disjoint ((𝓝 x).lift' closure) ((𝓝 y).lift' closure) ↔ x ≠ y | ⟨λ h hxy, by simpa [hxy, nhds_ne_bot.ne] using h, λ h, t2_5_space.t2_5 h⟩ | lemma | disjoint_lift'_closure_nhds | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closure",
"disjoint",
"t2_5_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t2_5_space.t2_space [t2_5_space α] : t2_space α | t2_space_iff_disjoint_nhds.2 $
λ x y hne, (disjoint_lift'_closure_nhds.2 hne).mono (le_lift'_closure _) (le_lift'_closure _) | instance | t2_5_space.t2_space | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"t2_5_space",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_nhds_disjoint_closure [t2_5_space α] {x y : α} (h : x ≠ y) :
∃ (s ∈ 𝓝 x) (t ∈ 𝓝 y), disjoint (closure s) (closure t) | ((𝓝 x).basis_sets.lift'_closure.disjoint_iff (𝓝 y).basis_sets.lift'_closure).1 $
disjoint_lift'_closure_nhds.2 h | lemma | exists_nhds_disjoint_closure | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closure",
"disjoint",
"t2_5_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_open_nhds_disjoint_closure [t2_5_space α] {x y : α} (h : x ≠ y) :
∃ u : set α, x ∈ u ∧ is_open u ∧ ∃ v : set α, y ∈ v ∧ is_open v ∧
disjoint (closure u) (closure v) | by simpa only [exists_prop, and.assoc] using ((nhds_basis_opens x).lift'_closure.disjoint_iff
(nhds_basis_opens y).lift'_closure).1 (disjoint_lift'_closure_nhds.2 h) | lemma | exists_open_nhds_disjoint_closure | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closure",
"disjoint",
"exists_prop",
"is_open",
"nhds_basis_opens",
"t2_5_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Lim_eq {a : α} [ne_bot f] (h : f ≤ 𝓝 a) :
@Lim _ _ ⟨a⟩ f = a | tendsto_nhds_unique (le_nhds_Lim ⟨a, h⟩) h | lemma | Lim_eq | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"Lim",
"le_nhds_Lim",
"tendsto_nhds_unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Lim_eq_iff [ne_bot f] (h : ∃ (a : α), f ≤ nhds a) {a} : @Lim _ _ ⟨a⟩ f = a ↔ f ≤ 𝓝 a | ⟨λ c, c ▸ le_nhds_Lim h, Lim_eq⟩ | lemma | Lim_eq_iff | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"Lim",
"le_nhds_Lim",
"nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ultrafilter.Lim_eq_iff_le_nhds [compact_space α] {x : α} {F : ultrafilter α} :
F.Lim = x ↔ ↑F ≤ 𝓝 x | ⟨λ h, h ▸ F.le_nhds_Lim, Lim_eq⟩ | lemma | ultrafilter.Lim_eq_iff_le_nhds | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"compact_space",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_iff_ultrafilter' [compact_space α] (U : set α) :
is_open U ↔ (∀ F : ultrafilter α, F.Lim ∈ U → U ∈ F.1) | begin
rw is_open_iff_ultrafilter,
refine ⟨λ h F hF, h F.Lim hF F F.le_nhds_Lim, _⟩,
intros cond x hx f h,
rw [← (ultrafilter.Lim_eq_iff_le_nhds.2 h)] at hx,
exact cond _ hx
end | lemma | is_open_iff_ultrafilter' | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"compact_space",
"is_open",
"is_open_iff_ultrafilter",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.lim_eq {a : α} {f : filter β} [ne_bot f] {g : β → α} (h : tendsto g f (𝓝 a)) :
@lim _ _ _ ⟨a⟩ f g = a | Lim_eq h | lemma | filter.tendsto.lim_eq | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"Lim_eq",
"filter",
"lim"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.lim_eq_iff {f : filter β} [ne_bot f] {g : β → α} (h : ∃ a, tendsto g f (𝓝 a)) {a} :
@lim _ _ _ ⟨a⟩ f g = a ↔ tendsto g f (𝓝 a) | ⟨λ c, c ▸ tendsto_nhds_lim h, filter.tendsto.lim_eq⟩ | lemma | filter.lim_eq_iff | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"filter",
"lim",
"tendsto_nhds_lim"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.lim_eq [topological_space β] {f : β → α} (h : continuous f) (a : β) :
@lim _ _ _ ⟨f a⟩ (𝓝 a) f = f a | (h.tendsto a).lim_eq | lemma | continuous.lim_eq | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"continuous",
"lim",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Lim_nhds (a : α) : @Lim _ _ ⟨a⟩ (𝓝 a) = a | Lim_eq le_rfl | lemma | Lim_nhds | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"Lim",
"Lim_eq",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lim_nhds_id (a : α) : @lim _ _ _ ⟨a⟩ (𝓝 a) id = a | Lim_nhds a | lemma | lim_nhds_id | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"Lim_nhds",
"lim"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Lim_nhds_within {a : α} {s : set α} (h : a ∈ closure s) :
@Lim _ _ ⟨a⟩ (𝓝[s] a) = a | by haveI : ne_bot (𝓝[s] a) := mem_closure_iff_cluster_pt.1 h;
exact Lim_eq inf_le_left | lemma | Lim_nhds_within | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"Lim",
"Lim_eq",
"closure",
"inf_le_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lim_nhds_within_id {a : α} {s : set α} (h : a ∈ closure s) :
@lim _ _ _ ⟨a⟩ (𝓝[s] a) id = a | Lim_nhds_within h | lemma | lim_nhds_within_id | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"Lim_nhds_within",
"closure",
"lim"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
discrete_topology.to_t2_space {α : Type*} [topological_space α] [discrete_topology α] :
t2_space α | ⟨λ x y h, ⟨{x}, {y}, is_open_discrete _, is_open_discrete _, rfl, rfl, disjoint_singleton.2 h⟩⟩ | instance | discrete_topology.to_t2_space | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"discrete_topology",
"is_open_discrete",
"t2_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
separated_by_continuous {α : Type*} {β : Type*}
[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 ∧ disjoint u v | let ⟨u, v, uo, vo, xu, yv, uv⟩ := t2_separation h in
⟨f ⁻¹' u, f ⁻¹' v, uo.preimage hf, vo.preimage hf, xu, yv, uv.preimage _⟩ | lemma | separated_by_continuous | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"continuous",
"disjoint",
"is_open",
"t2_separation",
"t2_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
separated_by_open_embedding {α β : Type*} [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 ∧ disjoint u v | let ⟨u, v, uo, vo, xu, yv, uv⟩ := t2_separation h in
⟨f '' u, f '' v, hf.is_open_map _ uo, hf.is_open_map _ vo,
mem_image_of_mem _ xu, mem_image_of_mem _ yv, disjoint_image_of_injective hf.inj uv⟩ | lemma | separated_by_open_embedding | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"disjoint",
"is_open",
"open_embedding",
"t2_separation",
"t2_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding.t2_space [topological_space β] [t2_space β] {f : α → β} (hf : embedding f) :
t2_space α | ⟨λ x y h, separated_by_continuous hf.continuous (hf.inj.ne h)⟩ | lemma | embedding.t2_space | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"embedding",
"separated_by_continuous",
"t2_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Pi.t2_space {α : Type*} {β : α → Type v} [t₂ : Πa, topological_space (β a)]
[∀a, t2_space (β a)] :
t2_space (Πa, β a) | ⟨assume x y h,
let ⟨i, hi⟩ := not_forall.mp (mt funext h) in
separated_by_continuous (continuous_apply i) hi⟩ | instance | Pi.t2_space | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"continuous_apply",
"separated_by_continuous",
"t2_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sigma.t2_space {ι : Type*} {α : ι → Type*} [Πi, topological_space (α i)]
[∀a, t2_space (α a)] :
t2_space (Σi, α i) | begin
constructor,
rintros ⟨i, x⟩ ⟨j, y⟩ neq,
rcases em (i = j) with (rfl|h),
{ replace neq : x ≠ y := λ c, (c.subst neq) rfl,
exact separated_by_open_embedding open_embedding_sigma_mk neq },
{ exact ⟨_, _, is_open_range_sigma_mk, is_open_range_sigma_mk, ⟨x, rfl⟩, ⟨y, rfl⟩,
set.disjoint_left.mpr $ b... | instance | sigma.t2_space | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"em",
"is_open_range_sigma_mk",
"open_embedding_sigma_mk",
"separated_by_open_embedding",
"t2_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_eq [t2_space α] {f g : β → α}
(hf : continuous f) (hg : continuous g) : is_closed {x:β | f x = g x} | continuous_iff_is_closed.mp (hf.prod_mk hg) _ is_closed_diagonal | lemma | is_closed_eq | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"continuous",
"is_closed",
"is_closed_diagonal",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_ne_fun [t2_space α] {f g : β → α}
(hf : continuous f) (hg : continuous g) : is_open {x:β | f x ≠ g x} | is_open_compl_iff.mpr $ is_closed_eq hf hg | lemma | is_open_ne_fun | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"continuous",
"is_closed_eq",
"is_open",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.eq_on.closure [t2_space α] {s : set β} {f g : β → α} (h : eq_on f g s)
(hf : continuous f) (hg : continuous g) :
eq_on f g (closure s) | closure_minimal h (is_closed_eq hf hg) | lemma | set.eq_on.closure | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closure",
"closure_minimal",
"continuous",
"is_closed_eq",
"t2_space"
] | If two continuous maps are equal on `s`, then they are equal on the closure of `s`. See also
`set.eq_on.of_subset_closure` for a more general version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous.ext_on [t2_space α] {s : set β} (hs : dense s) {f g : β → α}
(hf : continuous f) (hg : continuous g) (h : eq_on f g s) :
f = g | funext $ λ x, h.closure hf hg (hs x) | lemma | continuous.ext_on | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"continuous",
"dense",
"t2_space"
] | If two continuous functions are equal on a dense set, then they are equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_on_closure₂' [t2_space α] {s : set β} {t : set γ} {f g : β → γ → α}
(h : ∀ (x ∈ s) (y ∈ t), f x y = g x y)
(hf₁ : ∀ x, continuous (f x)) (hf₂ : ∀ y, continuous (λ x, f x y))
(hg₁ : ∀ x, continuous (g x)) (hg₂ : ∀ y, continuous (λ x, g x y)) :
∀ (x ∈ closure s) (y ∈ closure t), f x y = g x y | suffices closure s ⊆ ⋂ y ∈ closure t, {x | f x y = g x y}, by simpa only [subset_def, mem_Inter],
closure_minimal (λ x hx, mem_Inter₂.2 $ set.eq_on.closure (h x hx) (hf₁ _) (hg₁ _)) $
is_closed_bInter $ λ y hy, is_closed_eq (hf₂ _) (hg₂ _) | lemma | eq_on_closure₂' | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closure",
"closure_minimal",
"continuous",
"is_closed_bInter",
"is_closed_eq",
"set.eq_on.closure",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_on_closure₂ [t2_space α] {s : set β} {t : set γ} {f g : β → γ → α}
(h : ∀ (x ∈ s) (y ∈ t), f x y = g x y)
(hf : continuous (uncurry f)) (hg : continuous (uncurry g)) :
∀ (x ∈ closure s) (y ∈ closure t), f x y = g x y | eq_on_closure₂' h (λ x, continuous_uncurry_left x hf) (λ x, continuous_uncurry_right x hf)
(λ y, continuous_uncurry_left y hg) (λ y, continuous_uncurry_right y hg) | lemma | eq_on_closure₂ | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closure",
"continuous",
"continuous_uncurry_left",
"continuous_uncurry_right",
"eq_on_closure₂'",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.eq_on.of_subset_closure [t2_space α] {s t : set β} {f g : β → α} (h : eq_on f g s)
(hf : continuous_on f t) (hg : continuous_on g t) (hst : s ⊆ t) (hts : t ⊆ closure s) :
eq_on f g t | begin
intros x hx,
haveI : (𝓝[s] x).ne_bot, from mem_closure_iff_cluster_pt.mp (hts hx),
exact tendsto_nhds_unique_of_eventually_eq ((hf x hx).mono_left $ nhds_within_mono _ hst)
((hg x hx).mono_left $ nhds_within_mono _ hst) (h.eventually_eq_of_mem self_mem_nhds_within)
end | lemma | set.eq_on.of_subset_closure | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closure",
"continuous_on",
"nhds_within_mono",
"self_mem_nhds_within",
"t2_space",
"tendsto_nhds_unique_of_eventually_eq"
] | If `f x = g x` for all `x ∈ s` and `f`, `g` are continuous on `t`, `s ⊆ t ⊆ closure s`, then
`f x = g x` for all `x ∈ t`. See also `set.eq_on.closure`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.left_inverse.closed_range [t2_space α] {f : α → β} {g : β → α}
(h : function.left_inverse f g) (hf : continuous f) (hg : continuous g) :
is_closed (range g) | have eq_on (g ∘ f) id (closure $ range g),
from h.right_inv_on_range.eq_on.closure (hg.comp hf) continuous_id,
is_closed_of_closure_subset $ λ x hx,
calc x = g (f x) : (this hx).symm
... ∈ _ : mem_range_self _ | lemma | function.left_inverse.closed_range | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closure",
"continuous",
"continuous_id",
"is_closed",
"is_closed_of_closure_subset",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.left_inverse.closed_embedding [t2_space α] {f : α → β} {g : β → α}
(h : function.left_inverse f g) (hf : continuous f) (hg : continuous g) :
closed_embedding g | ⟨h.embedding hf hg, h.closed_range hf hg⟩ | lemma | function.left_inverse.closed_embedding | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closed_embedding",
"continuous",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_is_compact_separated [t2_space α] {s t : set α}
(hs : is_compact s) (ht : is_compact t) (hst : disjoint s t) :
separated_nhds s t | by simp only [separated_nhds, prod_subset_compl_diagonal_iff_disjoint.symm] at ⊢ hst;
exact generalized_tube_lemma hs ht is_closed_diagonal.is_open_compl hst | lemma | is_compact_is_compact_separated | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"disjoint",
"generalized_tube_lemma",
"is_compact",
"separated_nhds",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.is_closed [t2_space α] {s : set α} (hs : is_compact s) : is_closed s | is_open_compl_iff.1 $ is_open_iff_forall_mem_open.mpr $ assume x hx,
let ⟨u, v, uo, vo, su, xv, uv⟩ :=
is_compact_is_compact_separated hs is_compact_singleton (disjoint_singleton_right.2 hx) in
⟨v, (uv.mono_left $ show s ≤ u, from su).subset_compl_left, vo, by simpa using xv⟩ | lemma | is_compact.is_closed | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"is_closed",
"is_compact",
"is_compact_is_compact_separated",
"is_compact_singleton",
"t2_space"
] | In a `t2_space`, every compact set is closed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.coclosed_compact_eq_cocompact [t2_space α] :
coclosed_compact α = cocompact α | by simp [coclosed_compact, cocompact, infi_and', and_iff_right_of_imp is_compact.is_closed] | lemma | filter.coclosed_compact_eq_cocompact | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"and_iff_right_of_imp",
"infi_and'",
"is_compact.is_closed",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bornology.relatively_compact_eq_in_compact [t2_space α] :
bornology.relatively_compact α = bornology.in_compact α | by rw bornology.ext_iff; exact filter.coclosed_compact_eq_cocompact | lemma | bornology.relatively_compact_eq_in_compact | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"bornology.in_compact",
"bornology.relatively_compact",
"filter.coclosed_compact_eq_cocompact",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_subset_nhds_of_is_compact [t2_space α] {ι : Type*} [nonempty ι] {V : ι → set α}
(hV : directed (⊇) V) (hV_cpct : ∀ i, is_compact (V i)) {U : set α}
(hU : ∀ x ∈ ⋂ i, V i, U ∈ 𝓝 x) : ∃ i, V i ⊆ U | exists_subset_nhds_of_is_compact' hV hV_cpct (λ i, (hV_cpct i).is_closed) hU | lemma | exists_subset_nhds_of_is_compact | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"directed",
"exists_subset_nhds_of_is_compact'",
"is_closed",
"is_compact",
"t2_space"
] | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compact_exhaustion.is_closed [t2_space α] (K : compact_exhaustion α) (n : ℕ) :
is_closed (K n) | (K.is_compact n).is_closed | lemma | compact_exhaustion.is_closed | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"compact_exhaustion",
"is_closed",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.inter [t2_space α] {s t : set α} (hs : is_compact s) (ht : is_compact t) :
is_compact (s ∩ t) | hs.inter_right $ ht.is_closed | lemma | is_compact.inter | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"is_compact",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_closure_of_subset_compact [t2_space α] {s t : set α}
(ht : is_compact t) (h : s ⊆ t) : is_compact (closure s) | is_compact_of_is_closed_subset ht is_closed_closure (closure_minimal h ht.is_closed) | lemma | is_compact_closure_of_subset_compact | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closure",
"closure_minimal",
"is_closed_closure",
"is_compact",
"is_compact_of_is_closed_subset",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_compact_superset_iff [t2_space α] {s : set α} :
(∃ K, is_compact K ∧ s ⊆ K) ↔ is_compact (closure s) | ⟨λ ⟨K, hK, hsK⟩, is_compact_closure_of_subset_compact hK hsK, λ h, ⟨closure s, h, subset_closure⟩⟩ | lemma | exists_compact_superset_iff | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closure",
"is_compact",
"is_compact_closure_of_subset_compact",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_closure_of_is_compact [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) | subset.antisymm hf.image_closure $ closure_minimal (image_subset f subset_closure)
(hs.image_of_continuous_on hf).is_closed | lemma | image_closure_of_is_compact | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closure",
"closure_minimal",
"continuous_on",
"is_closed",
"is_compact",
"subset_closure",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.binary_compact_cover [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 | begin
obtain ⟨O₁, O₂, h1O₁, h1O₂, h2O₁, h2O₂, hO⟩ :=
is_compact_is_compact_separated (hK.diff hU) (hK.diff hV)
(by rwa [disjoint_iff_inter_eq_empty, diff_inter_diff, diff_eq_empty]),
exact ⟨_, _, hK.diff h1O₁, hK.diff h1O₂, by rwa [diff_subset_comm], by rwa [diff_subset_comm],
by rw [← diff_inter, hO.in... | lemma | is_compact.binary_compact_cover | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"is_compact",
"is_compact_is_compact_separated",
"is_open",
"t2_space"
] | If a compact set is covered by two open sets, then we can cover it by two compact subsets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous.is_closed_map [compact_space α] [t2_space β] {f : α → β}
(h : continuous f) : is_closed_map f | λ s hs, (hs.is_compact.image h).is_closed | lemma | continuous.is_closed_map | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"compact_space",
"continuous",
"is_closed",
"is_closed_map",
"t2_space"
] | A continuous map from a compact space to a Hausdorff space is a closed map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous.closed_embedding [compact_space α] [t2_space β] {f : α → β} (h : continuous f)
(hf : function.injective f) : closed_embedding f | closed_embedding_of_continuous_injective_closed h hf h.is_closed_map | lemma | continuous.closed_embedding | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closed_embedding",
"closed_embedding_of_continuous_injective_closed",
"compact_space",
"continuous",
"t2_space"
] | An injective continuous map from a compact space to a Hausdorff space is a closed embedding. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_map.of_surjective_continuous [compact_space α] [t2_space β] {f : α → β}
(hsurj : surjective f) (hcont : continuous f) : quotient_map f | hcont.is_closed_map.to_quotient_map hcont hsurj | lemma | quotient_map.of_surjective_continuous | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"compact_space",
"continuous",
"quotient_map",
"t2_space"
] | A surjective continuous map from a compact space to a Hausdorff space is a quotient map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact.finite_compact_cover [t2_space α] {s : set α} (hs : is_compact s)
{ι} (t : finset ι) (U : ι → set α) (hU : ∀ i ∈ t, is_open (U i)) (hsC : s ⊆ ⋃ i ∈ t, U i) :
∃ K : ι → set α, (∀ i, is_compact (K i)) ∧ (∀i, K i ⊆ U i) ∧ s = ⋃ i ∈ t, K i | begin
classical,
induction t using finset.induction with x t hx ih generalizing U hU s hs hsC,
{ refine ⟨λ _, ∅, λ i, is_compact_empty, λ i, empty_subset _, _⟩,
simpa only [subset_empty_iff, Union_false, Union_empty] using hsC },
simp only [finset.set_bUnion_insert] at hsC,
simp only [finset.mem_insert] a... | lemma | is_compact.finite_compact_cover | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"finset",
"finset.induction",
"finset.mem_insert",
"finset.set_bUnion_insert",
"ih",
"is_compact",
"is_compact_empty",
"is_open",
"is_open_bUnion",
"t2_space",
"update_noteq",
"update_same"
] | For every finite open cover `Uᵢ` of a compact set, there exists a compact cover `Kᵢ ⊆ Uᵢ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
locally_compact_of_compact_nhds [t2_space α] (h : ∀ x : α, ∃ s, s ∈ 𝓝 x ∧ is_compact s) :
locally_compact_space α | ⟨assume x n hn,
let ⟨u, un, uo, xu⟩ := mem_nhds_iff.mp hn in
let ⟨k, kx, kc⟩ := h x in
-- K is compact but not necessarily contained in N.
-- K \ U is again compact and doesn't contain x, so
-- we may find open sets V, W separating x from K \ U.
-- Then K \ W is a compact neighborhood of x contained in U.
... | lemma | locally_compact_of_compact_nhds | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"filter.inter_mem",
"is_compact",
"is_compact_is_compact_separated",
"is_compact_singleton",
"locally_compact_space",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_compact_of_compact [t2_space α] [compact_space α] : locally_compact_space α | locally_compact_of_compact_nhds (assume x, ⟨univ, is_open_univ.mem_nhds trivial, is_compact_univ⟩) | instance | locally_compact_of_compact | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"compact_space",
"locally_compact_of_compact_nhds",
"locally_compact_space",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_open_with_compact_closure [locally_compact_space α] [t2_space α] (x : α) :
∃ (U : set α), is_open U ∧ x ∈ U ∧ is_compact (closure U) | begin
rcases exists_compact_mem_nhds x with ⟨K, hKc, hxK⟩,
rcases mem_nhds_iff.1 hxK with ⟨t, h1t, h2t, h3t⟩,
exact ⟨t, h2t, h3t, is_compact_closure_of_subset_compact hKc h1t⟩
end | lemma | exists_open_with_compact_closure | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closure",
"exists_compact_mem_nhds",
"is_compact",
"is_compact_closure_of_subset_compact",
"is_open",
"locally_compact_space",
"t2_space"
] | In a locally compact T₂ space, every point has an open neighborhood with compact closure | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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