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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exists_open_superset_and_is_compact_closure [locally_compact_space α] [t2_space α]
{K : set α} (hK : is_compact K) : ∃ V, is_open V ∧ K ⊆ V ∧ is_compact (closure V) | begin
rcases exists_compact_superset hK with ⟨K', hK', hKK'⟩,
refine ⟨interior K', is_open_interior, hKK',
is_compact_closure_of_subset_compact hK' interior_subset⟩,
end | lemma | exists_open_superset_and_is_compact_closure | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closure",
"exists_compact_superset",
"is_compact",
"is_compact_closure_of_subset_compact",
"is_open",
"is_open_interior",
"locally_compact_space",
"t2_space"
] | In a locally compact T₂ space, every compact set has an open neighborhood with compact closure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_open_between_and_is_compact_closure [locally_compact_space α] [t2_space α]
{K U : set α} (hK : is_compact K) (hU : is_open U) (hKU : K ⊆ U) :
∃ V, is_open V ∧ K ⊆ V ∧ closure V ⊆ U ∧ is_compact (closure V) | begin
rcases exists_compact_between hK hU hKU with ⟨V, hV, hKV, hVU⟩,
exact ⟨interior V, is_open_interior, hKV,
(closure_minimal interior_subset hV.is_closed).trans hVU,
is_compact_closure_of_subset_compact hV interior_subset⟩,
end | lemma | exists_open_between_and_is_compact_closure | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closure",
"closure_minimal",
"exists_compact_between",
"interior_subset",
"is_compact",
"is_compact_closure_of_subset_compact",
"is_open",
"is_open_interior",
"locally_compact_space",
"t2_space"
] | In a locally compact T₂ 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`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_preirreducible_iff_subsingleton [t2_space α] {S : set α} :
is_preirreducible S ↔ S.subsingleton | begin
refine ⟨λ h x hx y hy, _, set.subsingleton.is_preirreducible⟩,
by_contradiction e,
obtain ⟨U, V, hU, hV, hxU, hyV, h'⟩ := t2_separation e,
exact ((h U V hU hV ⟨x, hx, hxU⟩ ⟨y, hy, hyV⟩).mono $ inter_subset_right _ _).not_disjoint h',
end | lemma | is_preirreducible_iff_subsingleton | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"by_contradiction",
"is_preirreducible",
"t2_separation",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_irreducible_iff_singleton [t2_space α] {S : set α} :
is_irreducible S ↔ ∃ x, S = {x} | by rw [is_irreducible, is_preirreducible_iff_subsingleton,
exists_eq_singleton_iff_nonempty_subsingleton] | lemma | is_irreducible_iff_singleton | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"is_irreducible",
"is_preirreducible_iff_subsingleton",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_preirreducible_nontrivial_t2 (α) [topological_space α] [preirreducible_space α]
[nontrivial α] [t2_space α] : false | (preirreducible_space.is_preirreducible_univ α).subsingleton.not_nontrivial nontrivial_univ | lemma | not_preirreducible_nontrivial_t2 | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"nontrivial",
"preirreducible_space",
"t2_space",
"topological_space"
] | There does not exist a nontrivial preirreducible T₂ space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
regular_space (X : Type u) [topological_space X] : Prop | (regular : ∀ {s : set X} {a}, is_closed s → a ∉ s → disjoint (𝓝ˢ s) (𝓝 a)) | class | regular_space | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"disjoint",
"is_closed",
"topological_space"
] | A topological space is called a *regular space* if for any closed set `s` and `a ∉ s`, there
exist disjoint open sets `U ⊇ s` and `V ∋ a`. We formulate this condition in terms of `disjoint`ness
of filters `𝓝ˢ s` and `𝓝 a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
regular_space_tfae (X : Type u) [topological_space X] :
tfae [regular_space X,
∀ (s : set X) (a ∉ closure s), disjoint (𝓝ˢ s) (𝓝 a),
∀ (a : X) (s : set X), disjoint (𝓝ˢ s) (𝓝 a) ↔ a ∉ closure s,
∀ (a : X) (s ∈ 𝓝 a), ∃ t ∈ 𝓝 a, is_closed t ∧ t ⊆ s,
∀ a : X, (𝓝 a).lift' closure ≤ 𝓝 a,
∀ a : ... | begin
tfae_have : 1 ↔ 5,
{ rw [regular_space_iff, (@compl_surjective (set X) _).forall, forall_swap],
simp only [is_closed_compl_iff, mem_compl_iff, not_not, @and_comm (_ ∈ _),
(nhds_basis_opens _).lift'_closure.le_basis_iff (nhds_basis_opens _), and_imp,
(nhds_basis_opens _).disjoint_iff_right, exi... | lemma | regular_space_tfae | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"and_imp",
"closure",
"compl_surjective",
"disjoint",
"disjoint_compl_left",
"exists_prop",
"forall_swap",
"interior_compl",
"is_closed",
"is_closed_closure",
"is_closed_compl_iff",
"mem_interior_iff_mem_nhds",
"nhds_basis_opens",
"not_not",
"principal_le_nhds_set",
"regular_space",
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
regular_space.of_lift'_closure (h : ∀ a : α, (𝓝 a).lift' closure = 𝓝 a) : regular_space α | iff.mpr ((regular_space_tfae α).out 0 5) h | lemma | regular_space.of_lift'_closure | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closure",
"regular_space",
"regular_space_tfae"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
regular_space.of_basis {ι : α → Sort*} {p : Π a, ι a → Prop} {s : Π a, ι a → set α}
(h₁ : ∀ a, (𝓝 a).has_basis (p a) (s a)) (h₂ : ∀ a i, p a i → is_closed (s a i)) :
regular_space α | regular_space.of_lift'_closure $ λ a, (h₁ a).lift'_closure_eq_self (h₂ a) | lemma | regular_space.of_basis | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"is_closed",
"regular_space",
"regular_space.of_lift'_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
regular_space.of_exists_mem_nhds_is_closed_subset
(h : ∀ (a : α) (s ∈ 𝓝 a), ∃ t ∈ 𝓝 a, is_closed t ∧ t ⊆ s) : regular_space α | iff.mpr ((regular_space_tfae α).out 0 3) h | lemma | regular_space.of_exists_mem_nhds_is_closed_subset | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"is_closed",
"regular_space",
"regular_space_tfae"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_nhds_set_nhds : disjoint (𝓝ˢ s) (𝓝 a) ↔ a ∉ closure s | iff.mp ((regular_space_tfae α).out 0 2) ‹_› _ _ | lemma | disjoint_nhds_set_nhds | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closure",
"disjoint",
"regular_space_tfae"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_nhds_nhds_set : disjoint (𝓝 a) (𝓝ˢ s) ↔ a ∉ closure s | disjoint.comm.trans disjoint_nhds_set_nhds | lemma | disjoint_nhds_nhds_set | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closure",
"disjoint",
"disjoint_nhds_set_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_mem_nhds_is_closed_subset {a : α} {s : set α} (h : s ∈ 𝓝 a) :
∃ t ∈ 𝓝 a, is_closed t ∧ t ⊆ s | iff.mp ((regular_space_tfae α).out 0 3) ‹_› _ _ h | lemma | exists_mem_nhds_is_closed_subset | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"is_closed",
"regular_space_tfae"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_nhds_basis (a : α) : (𝓝 a).has_basis (λ s : set α, s ∈ 𝓝 a ∧ is_closed s) id | has_basis_self.2 (λ _, exists_mem_nhds_is_closed_subset) | lemma | closed_nhds_basis | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"exists_mem_nhds_is_closed_subset",
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift'_nhds_closure (a : α) : (𝓝 a).lift' closure = 𝓝 a | (closed_nhds_basis a).lift'_closure_eq_self (λ s hs, hs.2) | lemma | lift'_nhds_closure | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closed_nhds_basis",
"closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.has_basis.nhds_closure {ι : Sort*} {a : α} {p : ι → Prop} {s : ι → set α}
(h : (𝓝 a).has_basis p s) : (𝓝 a).has_basis p (λ i, closure (s i)) | lift'_nhds_closure a ▸ h.lift'_closure | lemma | filter.has_basis.nhds_closure | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closure",
"lift'_nhds_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_basis_nhds_closure (a : α) : (𝓝 a).has_basis (λ s, s ∈ 𝓝 a) closure | (𝓝 a).basis_sets.nhds_closure | lemma | has_basis_nhds_closure | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_basis_opens_closure (a : α) : (𝓝 a).has_basis (λ s, a ∈ s ∧ is_open s) closure | (nhds_basis_opens a).nhds_closure | lemma | has_basis_opens_closure | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closure",
"is_open",
"nhds_basis_opens"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_space.is_topological_basis.nhds_basis_closure
{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 | by simpa only [and_comm] using hB.nhds_has_basis.nhds_closure | lemma | topological_space.is_topological_basis.nhds_basis_closure | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closure",
"topological_space.is_topological_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_space.is_topological_basis.exists_closure_subset
{B : set (set α)} (hB : topological_space.is_topological_basis B) {a : α} {s : set α}
(h : s ∈ 𝓝 a) :
∃ t ∈ B, a ∈ t ∧ closure t ⊆ s | by simpa only [exists_prop, and.assoc] using hB.nhds_has_basis.nhds_closure.mem_iff.mp h | lemma | topological_space.is_topological_basis.exists_closure_subset | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closure",
"exists_prop",
"topological_space.is_topological_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_nhds_nhds_iff_not_specializes {a b : α} :
disjoint (𝓝 a) (𝓝 b) ↔ ¬a ⤳ b | by rw [← nhds_set_singleton, disjoint_nhds_set_nhds, specializes_iff_mem_closure] | lemma | disjoint_nhds_nhds_iff_not_specializes | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"disjoint",
"disjoint_nhds_set_nhds",
"nhds_set_singleton",
"specializes_iff_mem_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
specializes_comm {a b : α} : a ⤳ b ↔ b ⤳ a | by simp only [← disjoint_nhds_nhds_iff_not_specializes.not_left, disjoint.comm] | lemma | specializes_comm | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"disjoint.comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
specializes_iff_inseparable {a b : α} : a ⤳ b ↔ inseparable a b | ⟨λ h, h.antisymm h.symm, le_of_eq⟩ | lemma | specializes_iff_inseparable | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"inseparable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_set_of_specializes : is_closed {p : α × α | p.1 ⤳ p.2} | by simp only [← is_open_compl_iff, compl_set_of, ← disjoint_nhds_nhds_iff_not_specializes,
is_open_set_of_disjoint_nhds_nhds] | lemma | is_closed_set_of_specializes | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"disjoint_nhds_nhds_iff_not_specializes",
"is_closed",
"is_open_compl_iff",
"is_open_set_of_disjoint_nhds_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_set_of_inseparable : is_closed {p : α × α | inseparable p.1 p.2} | by simp only [← specializes_iff_inseparable, is_closed_set_of_specializes] | lemma | is_closed_set_of_inseparable | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"inseparable",
"is_closed",
"is_closed_set_of_specializes",
"specializes_iff_inseparable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.regular_space [topological_space β] {f : β → α} (hf : inducing f) :
regular_space β | regular_space.of_basis (λ b, by { rw [hf.nhds_eq_comap b], exact (closed_nhds_basis _).comap _ }) $
λ b s hs, hs.2.preimage hf.continuous | lemma | inducing.regular_space | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closed_nhds_basis",
"inducing",
"regular_space",
"regular_space.of_basis",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
regular_space_induced (f : β → α) : @regular_space β (induced f ‹_›) | by { letI := induced f ‹_›, exact inducing.regular_space ⟨rfl⟩ } | lemma | regular_space_induced | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"inducing.regular_space",
"regular_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
regular_space_Inf {X} {T : set (topological_space X)} (h : ∀ t ∈ T, @regular_space X t) :
@regular_space X (Inf T) | begin
letI := Inf T,
have : ∀ a, (𝓝 a).has_basis
(λ If : Σ I : set T, I → set X,
If.1.finite ∧ ∀ i : If.1, If.2 i ∈ @nhds X i a ∧ is_closed[↑i] (If.2 i))
(λ If, ⋂ i : If.1, If.snd i),
{ intro a,
rw [nhds_Inf, ← infi_subtype''],
exact has_basis_infi (λ t : T, @closed_nhds_basis X t (h t t.2)... | lemma | regular_space_Inf | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"Inf_le",
"closed_nhds_basis",
"infi_subtype''",
"is_closed",
"is_closed_Inter",
"nhds",
"nhds_Inf",
"regular_space",
"regular_space.of_basis",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
regular_space_infi {ι X} {t : ι → topological_space X} (h : ∀ i, @regular_space X (t i)) :
@regular_space X (infi t) | regular_space_Inf $ forall_range_iff.mpr h | lemma | regular_space_infi | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"infi",
"regular_space",
"regular_space_Inf",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
regular_space.inf {X} {t₁ t₂ : topological_space X} (h₁ : @regular_space X t₁)
(h₂ : @regular_space X t₂) : @regular_space X (t₁ ⊓ t₂) | by { rw [inf_eq_infi], exact regular_space_infi (bool.forall_bool.2 ⟨h₂, h₁⟩) } | lemma | regular_space.inf | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"inf_eq_infi",
"regular_space",
"regular_space_infi",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t3_space (α : Type u) [topological_space α] extends t0_space α, regular_space α : Prop | class | t3_space | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"regular_space",
"t0_space",
"topological_space"
] | A T₃ space is a T₀ space which is a regular space. Any T₃ space is a T₁ space, a T₂ space, and
a T₂.₅ space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t3_space.t2_5_space [t3_space α] : t2_5_space α | begin
refine ⟨λ x y hne, _⟩,
rw [lift'_nhds_closure, lift'_nhds_closure],
have aux : x ∉ closure {y} ∨ y ∉ closure {x},
from (t0_space_iff_or_not_mem_closure α).mp infer_instance x y hne,
wlog H : x ∉ closure ({y} : set α),
{ refine (this y x hne.symm aux.symm (aux.resolve_left H)).symm },
{ rwa [← disj... | instance | t3_space.t2_5_space | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"aux",
"closure",
"disjoint_nhds_nhds_set",
"lift'_nhds_closure",
"nhds_set_singleton",
"t0_space_iff_or_not_mem_closure",
"t2_5_space",
"t3_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding.t3_space [topological_space β] [t3_space β] {f : α → β}
(hf : embedding f) : t3_space α | { to_t0_space := hf.t0_space,
to_regular_space := hf.to_inducing.regular_space } | lemma | embedding.t3_space | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"embedding",
"t3_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype.t3_space [t3_space α] {p : α → Prop} : t3_space (subtype p) | embedding_subtype_coe.t3_space | instance | subtype.t3_space | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"t3_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_nested_nhds [t3_space α] {x y : α} (h : x ≠ y) :
∃ (U₁ V₁ ∈ 𝓝 x) (U₂ V₂ ∈ 𝓝 y), is_closed V₁ ∧ is_closed V₂ ∧ is_open U₁ ∧ is_open U₂ ∧
V₁ ⊆ U₁ ∧ V₂ ⊆ U₂ ∧ disjoint U₁ U₂ | begin
rcases t2_separation h with ⟨U₁, U₂, U₁_op, U₂_op, x_in, y_in, H⟩,
rcases exists_mem_nhds_is_closed_subset (U₁_op.mem_nhds x_in) with ⟨V₁, V₁_in, V₁_closed, h₁⟩,
rcases exists_mem_nhds_is_closed_subset (U₂_op.mem_nhds y_in) with ⟨V₂, V₂_in, V₂_closed, h₂⟩,
exact ⟨U₁, mem_of_superset V₁_in h₁, V₁, V₁_in, U... | lemma | disjoint_nested_nhds | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"disjoint",
"exists_mem_nhds_is_closed_subset",
"is_closed",
"is_open",
"t2_separation",
"t3_space"
] | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normal_space (α : Type u) [topological_space α] extends t1_space α : Prop | (normal : ∀ s t : set α, is_closed s → is_closed t → disjoint s t → separated_nhds s t) | class | normal_space | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"disjoint",
"is_closed",
"normal",
"separated_nhds",
"t1_space",
"topological_space"
] | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normal_separation [normal_space α] {s t : set α}
(H1 : is_closed s) (H2 : is_closed t) (H3 : disjoint s t) :
separated_nhds s t | normal_space.normal s t H1 H2 H3 | theorem | normal_separation | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"disjoint",
"is_closed",
"normal_space",
"separated_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normal_exists_closure_subset [normal_space α] {s t : set α} (hs : is_closed s)
(ht : is_open t) (hst : s ⊆ t) :
∃ u, is_open u ∧ s ⊆ u ∧ closure u ⊆ t | begin
have : disjoint s tᶜ, from set.disjoint_left.mpr (λ x hxs hxt, hxt (hst hxs)),
rcases normal_separation hs (is_closed_compl_iff.2 ht) this
with ⟨s', t', hs', ht', hss', htt', hs't'⟩,
refine ⟨s', hs', hss',
subset.trans (closure_minimal _ (is_closed_compl_iff.2 ht')) (compl_subset_comm.1 htt')⟩,
ex... | theorem | normal_exists_closure_subset | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closure",
"closure_minimal",
"disjoint",
"is_closed",
"is_open",
"normal_separation",
"normal_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normal_space.t3_space [normal_space α] : t3_space α | { regular := λ s x hs hxs, let ⟨u, v, hu, hv, hsu, hxv, huv⟩ :=
normal_separation hs is_closed_singleton (disjoint_singleton_right.mpr hxs) in
disjoint_of_disjoint_of_mem huv (hu.mem_nhds_set.2 hsu) (hv.mem_nhds $ hxv rfl) } | instance | normal_space.t3_space | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"is_closed_singleton",
"normal_separation",
"normal_space",
"t3_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normal_of_compact_t2 [compact_space α] [t2_space α] : normal_space α | ⟨λ s t hs ht, is_compact_is_compact_separated hs.is_compact ht.is_compact⟩ | lemma | normal_of_compact_t2 | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"compact_space",
"is_compact_is_compact_separated",
"normal_space",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_embedding.normal_space [topological_space β] [normal_space β] {f : α → β}
(hf : closed_embedding f) : normal_space α | { to_t1_space := hf.to_embedding.t1_space,
normal :=
begin
intros s t hs ht hst,
have H : separated_nhds (f '' s) (f '' t),
from normal_space.normal (f '' s) (f '' t) (hf.is_closed_map s hs) (hf.is_closed_map t ht)
(disjoint_image_of_injective hf.inj hst),
exact (H.preimage hf.continuous).... | lemma | closed_embedding.normal_space | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closed_embedding",
"normal",
"normal_space",
"separated_nhds",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normal_space_of_t3_second_countable [second_countable_topology α] [t3_space α] :
normal_space α | begin
have key : ∀ {s t : set α}, is_closed t → disjoint s t →
∃ U : set (countable_basis α), (s ⊆ ⋃ u ∈ U, ↑u) ∧
(∀ u ∈ U, disjoint (closure ↑u) t) ∧
∀ n : ℕ, is_closed (⋃ (u ∈ U) (h : encodable.encode u ≤ n), closure (u : set α)),
{ intros s t hc hd,
rw disjoint_left at hd,
have : ∀ x ∈ s,... | lemma | normal_space_of_t3_second_countable | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closure",
"disjoint",
"encodable.encode'",
"is_closed",
"is_closed_bUnion",
"is_closed_closure",
"is_open_bUnion",
"normal_space",
"not_and",
"not_exists",
"not_forall",
"not_not",
"subset_closure",
"supr_and'",
"t3_space"
] | A T₃ topological space with second countable topology is a normal space.
This lemma is not an instance to avoid a loop. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
t5_space (α : Type u) [topological_space α] extends t1_space α : Prop | (completely_normal : ∀ ⦃s t : set α⦄, disjoint (closure s) t → disjoint s (closure t) →
disjoint (𝓝ˢ s) (𝓝ˢ t)) | class | t5_space | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"closure",
"disjoint",
"t1_space",
"topological_space"
] | A topological space `α` is a *completely normal Hausdorff space* if each subspace `s : set α` is
a normal Hausdorff space. Equivalently, `α` is a `T₁` space and for any two sets `s`, `t` such that
`closure s` is disjoint with `t` and `s` is disjoint with `closure t`, there exist disjoint
neighbourhoods of `s` and `t`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
embedding.t5_space [topological_space β] [t5_space β] {e : α → β} (he : embedding e) :
t5_space α | begin
haveI := he.t1_space,
refine ⟨λ s t hd₁ hd₂, _⟩,
simp only [he.to_inducing.nhds_set_eq_comap],
refine disjoint_comap (completely_normal _ _),
{ rwa [← subset_compl_iff_disjoint_left, image_subset_iff, preimage_compl,
← he.closure_eq_preimage_closure_image, subset_compl_iff_disjoint_left] },
{ rw... | lemma | embedding.t5_space | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"embedding",
"t5_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t5_space.to_normal_space [t5_space α] : normal_space α | ⟨λ s t hs ht hd, separated_nhds_iff_disjoint.2 $
completely_normal (by rwa [hs.closure_eq]) (by rwa [ht.closure_eq])⟩ | instance | t5_space.to_normal_space | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"normal_space",
"t5_space"
] | A `T₅` space is a `T₄` space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
connected_component_eq_Inter_clopen [t2_space α] [compact_space α] (x : α) :
connected_component x = ⋂ Z : {Z : set α // is_clopen Z ∧ x ∈ Z}, Z | begin
apply eq_of_subset_of_subset connected_component_subset_Inter_clopen,
-- Reduce to showing that the clopen intersection is connected.
refine is_preconnected.subset_connected_component _ (mem_Inter.2 (λ Z, Z.2.2)),
-- We do this by showing that any disjoint cover by two closed sets implies
-- that one of... | lemma | connected_component_eq_Inter_clopen | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"compact_space",
"connected_component",
"connected_component_subset_Inter_clopen",
"disjoint.left_le_of_le_sup_right",
"imp_not_comm",
"is_clopen",
"is_clopen_bInter_finset",
"is_clopen_inter_of_disjoint_cover_clopen",
"is_closed",
"is_closed_Inter",
"is_preconnected.subset_connected_component",... | In a compact t2 space, the connected component of a point equals the intersection of all
its clopen neighbourhoods. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
totally_separated_space_of_t1_of_basis_clopen [t1_space α]
(h : is_topological_basis {s : set α | is_clopen s}) :
totally_separated_space α | begin
constructor,
rintros x - y - hxy,
rcases h.mem_nhds_iff.mp (is_open_ne.mem_nhds hxy) with ⟨U, hU, hxU, hyU⟩,
exact ⟨U, Uᶜ, hU.is_open, hU.compl.is_open, hxU, λ h, hyU h rfl,
(union_compl_self U).superset, disjoint_compl_right⟩
end | lemma | totally_separated_space_of_t1_of_basis_clopen | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"is_clopen",
"t1_space",
"totally_separated_space"
] | A T1 space with a clopen basis is totally separated. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compact_t2_tot_disc_iff_tot_sep :
totally_disconnected_space α ↔ totally_separated_space α | begin
split,
{ intro h, constructor,
rintros x - y -,
contrapose!,
intros hyp,
suffices : x ∈ connected_component y,
by simpa [totally_disconnected_space_iff_connected_component_singleton.1 h y,
mem_singleton_iff],
rw [connected_component_eq_Inter_clopen, mem_Inter],
ri... | theorem | compact_t2_tot_disc_iff_tot_sep | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"by_contra",
"connected_component",
"connected_component_eq_Inter_clopen",
"disjoint_compl_left",
"is_clopen",
"is_compl_compl",
"totally_disconnected_space",
"totally_separated_space",
"totally_separated_space.totally_disconnected_space"
] | A compact Hausdorff space is totally disconnected if and only if it is totally separated, this
is also true for locally compact spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nhds_basis_clopen (x : α) : (𝓝 x).has_basis (λ s : set α, x ∈ s ∧ is_clopen s) id | ⟨λ U, begin
split,
{ have : connected_component x = {x},
from totally_disconnected_space_iff_connected_component_singleton.mp ‹_› x,
rw connected_component_eq_Inter_clopen at this,
intros hU,
let N := {Z // is_clopen Z ∧ x ∈ Z},
rsuffices ⟨⟨s, hs, hs'⟩, hs''⟩ : ∃ Z : N, Z.val ⊆ U,
{ exact ... | lemma | nhds_basis_clopen | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"connected_component",
"connected_component_eq_Inter_clopen",
"directed",
"exists_subset_nhds_of_compact_space",
"is_clopen",
"is_clopen_univ",
"is_closed",
"mem_nhds_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_topological_basis_clopen : is_topological_basis {s : set α | is_clopen s} | begin
apply is_topological_basis_of_open_of_nhds (λ U (hU : is_clopen U), hU.1),
intros x U hxU U_op,
have : U ∈ 𝓝 x,
from is_open.mem_nhds U_op hxU,
rcases (nhds_basis_clopen x).mem_iff.mp this with ⟨V, ⟨hxV, hV⟩, hVU : V ⊆ U⟩,
use V,
tauto
end | lemma | is_topological_basis_clopen | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"is_clopen",
"is_open.mem_nhds",
"nhds_basis_clopen"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compact_exists_clopen_in_open {x : α} {U : set α} (is_open : is_open U) (memU : x ∈ U) :
∃ (V : set α) (hV : is_clopen V), x ∈ V ∧ V ⊆ U | (is_topological_basis.mem_nhds_iff is_topological_basis_clopen).1 (is_open.mem_nhds memU) | lemma | compact_exists_clopen_in_open | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"is_clopen",
"is_open",
"is_open.mem_nhds",
"is_topological_basis_clopen"
] | Every member of an open set in a compact Hausdorff totally disconnected space
is contained in a clopen set contained in the open set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
loc_compact_Haus_tot_disc_of_zero_dim [totally_disconnected_space H] :
is_topological_basis {s : set H | is_clopen s} | begin
refine is_topological_basis_of_open_of_nhds (λ u hu, hu.1) _,
rintros x U memU hU,
obtain ⟨s, comp, xs, sU⟩ := exists_compact_subset hU memU,
obtain ⟨t, h, ht, xt⟩ := mem_interior.1 xs,
let u : set s := (coe : s → H)⁻¹' (interior s),
have u_open_in_s : is_open u := is_open_interior.preimage continuous... | lemma | loc_compact_Haus_tot_disc_of_zero_dim | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"compact_exists_clopen_in_open",
"compact_space",
"continuous_subtype_coe",
"embedding",
"embedding_subtype_coe",
"exists_compact_subset",
"interior",
"interior_subset",
"is_clopen",
"is_open",
"is_open_interior",
"open_embedding",
"set.inter_eq_self_of_subset_left",
"set.range",
"set.ra... | A locally compact Hausdorff totally disconnected space has a basis with clopen elements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
loc_compact_t2_tot_disc_iff_tot_sep :
totally_disconnected_space H ↔ totally_separated_space H | begin
split,
{ introI h,
exact totally_separated_space_of_t1_of_basis_clopen loc_compact_Haus_tot_disc_of_zero_dim },
apply totally_separated_space.totally_disconnected_space,
end | theorem | loc_compact_t2_tot_disc_iff_tot_sep | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"loc_compact_Haus_tot_disc_of_zero_dim",
"totally_disconnected_space",
"totally_separated_space",
"totally_separated_space.totally_disconnected_space",
"totally_separated_space_of_t1_of_basis_clopen"
] | A locally compact Hausdorff space is totally disconnected
if and only if it is totally separated. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
connected_components.t2 [t2_space α] [compact_space α] :
t2_space (connected_components α) | begin
-- Proof follows that of: https://stacks.math.columbia.edu/tag/0900
-- Fix 2 distinct connected components, with points a and b
refine ⟨connected_components.surjective_coe.forall₂.2 $ λ a b ne, _⟩,
rw connected_components.coe_ne_coe at ne,
have h := connected_component_disjoint ne,
-- write ↑b as the ... | instance | connected_components.t2 | topology | src/topology/separation.lean | [
"topology.subset_properties",
"topology.connected",
"topology.nhds_set",
"topology.inseparable"
] | [
"compact_space",
"connected_component",
"connected_component_disjoint",
"connected_component_eq_Inter_clopen",
"connected_components",
"connected_components.coe_ne_coe",
"connected_components_preimage_image",
"is_clopen",
"is_clopen_bInter_finset",
"mem_connected_component",
"t2_space"
] | `connected_components α` is Hausdorff when `α` is Hausdorff and compact | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seq_closure (s : set X) : set X | {a | ∃ x : ℕ → X, (∀ n : ℕ, x n ∈ s) ∧ tendsto x at_top (𝓝 a)} | def | seq_closure | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [] | The sequential closure of a set `s : set X` in a topological space `X` is the set of all `a : X`
which arise as limit of sequences in `s`. Note that the sequential closure of a set is not
guaranteed to be sequentially closed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subset_seq_closure {s : set X} : s ⊆ seq_closure s | λ p hp, ⟨const ℕ p, λ _, hp, tendsto_const_nhds⟩ | lemma | subset_seq_closure | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"seq_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seq_closure_subset_closure {s : set X} : seq_closure s ⊆ closure s | λ p ⟨x, xM, xp⟩, mem_closure_of_tendsto xp (univ_mem' xM) | lemma | seq_closure_subset_closure | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"closure",
"mem_closure_of_tendsto",
"seq_closure"
] | The sequential closure of a set is contained in the closure of that set.
The converse is not true. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_seq_closed (s : set X) : Prop | ∀ ⦃x : ℕ → X⦄ ⦃p : X⦄, (∀ n, x n ∈ s) → tendsto x at_top (𝓝 p) → p ∈ s | def | is_seq_closed | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [] | A set `s` is sequentially closed if for any converging sequence `x n` of elements of `s`, the
limit belongs to `s` as well. Note that the sequential closure of a set is not guaranteed to be
sequentially closed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_seq_closed.seq_closure_eq {s : set X} (hs : is_seq_closed s) :
seq_closure s = s | subset.antisymm (λ p ⟨x, hx, hp⟩, hs hx hp) subset_seq_closure | lemma | is_seq_closed.seq_closure_eq | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"is_seq_closed",
"seq_closure",
"subset_seq_closure"
] | The sequential closure of a sequentially closed set is the set itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_seq_closed_of_seq_closure_eq {s : set X} (hs : seq_closure s = s) :
is_seq_closed s | λ x p hxs hxp, hs ▸ ⟨x, hxs, hxp⟩ | lemma | is_seq_closed_of_seq_closure_eq | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"is_seq_closed",
"seq_closure"
] | If a set is equal to its sequential closure, then it is sequentially closed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_seq_closed_iff {s : set X} :
is_seq_closed s ↔ seq_closure s = s | ⟨is_seq_closed.seq_closure_eq, is_seq_closed_of_seq_closure_eq⟩ | lemma | is_seq_closed_iff | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"is_seq_closed",
"seq_closure"
] | A set is sequentially closed iff it is equal to its sequential closure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closed.is_seq_closed {s : set X} (hc : is_closed s) : is_seq_closed s | λ u x hu hx, hc.mem_of_tendsto hx (eventually_of_forall hu) | lemma | is_closed.is_seq_closed | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"is_closed",
"is_seq_closed"
] | A set is sequentially closed if it is closed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
frechet_urysohn_space (X : Type*) [topological_space X] : Prop | (closure_subset_seq_closure : ∀ s : set X, closure s ⊆ seq_closure s) | class | frechet_urysohn_space | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"closure",
"seq_closure",
"topological_space"
] | A topological space is called a *Fréchet-Urysohn space*, if the sequential closure of any set
is equal to its closure. Since one of the inclusions is trivial, we require only the non-trivial one
in the definition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seq_closure_eq_closure [frechet_urysohn_space X] (s : set X) :
seq_closure s = closure s | seq_closure_subset_closure.antisymm $ frechet_urysohn_space.closure_subset_seq_closure s | lemma | seq_closure_eq_closure | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"closure",
"frechet_urysohn_space",
"seq_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closure_iff_seq_limit [frechet_urysohn_space X] {s : set X} {a : X} :
a ∈ closure s ↔ ∃ x : ℕ → X, (∀ n : ℕ, x n ∈ s) ∧ tendsto x at_top (𝓝 a) | by { rw [← seq_closure_eq_closure], refl } | lemma | mem_closure_iff_seq_limit | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"closure",
"frechet_urysohn_space",
"seq_closure_eq_closure"
] | In a Fréchet-Urysohn space, a point belongs to the closure of a set iff it is a limit
of a sequence taking values in this set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_nhds_iff_seq_tendsto [frechet_urysohn_space X] {f : X → Y} {a : X} {b : Y} :
tendsto f (𝓝 a) (𝓝 b) ↔ ∀ u : ℕ → X, tendsto u at_top (𝓝 a) → tendsto (f ∘ u) at_top (𝓝 b) | begin
refine ⟨λ hf u hu, hf.comp hu,
λ h, ((nhds_basis_closeds _).tendsto_iff (nhds_basis_closeds _)).2 _⟩,
rintro s ⟨hbs, hsc⟩,
refine ⟨closure (f ⁻¹' s), ⟨mt _ hbs, is_closed_closure⟩, λ x, mt $ λ hx, subset_closure hx⟩,
rw [← seq_closure_eq_closure],
rintro ⟨u, hus, hu⟩,
exact hsc.mem_of_tendsto (h u... | lemma | tendsto_nhds_iff_seq_tendsto | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"frechet_urysohn_space",
"nhds_basis_closeds",
"seq_closure_eq_closure",
"subset_closure"
] | If the domain of a function `f : α → β` is a Fréchet-Urysohn space, then convergence
is equivalent to sequential convergence. See also `filter.tendsto_iff_seq_tendsto` for a version
that works for any pair of filters assuming that the filter in the domain is countably generated.
This property is equivalent to the defi... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
frechet_urysohn_space.of_seq_tendsto_imp_tendsto
(h : ∀ (f : X → Prop) (a : X),
(∀ u : ℕ → X, tendsto u at_top (𝓝 a) → tendsto (f ∘ u) at_top (𝓝 (f a))) → continuous_at f a) :
frechet_urysohn_space X | begin
refine ⟨λ s x hcx, _⟩,
specialize h (∉ s) x,
by_cases hx : x ∈ s, { exact subset_seq_closure hx },
simp_rw [(∘), continuous_at, hx, not_false_iff, nhds_true, tendsto_pure, eq_true_iff,
← mem_compl_iff, eventually_mem_set, ← mem_interior_iff_mem_nhds, interior_compl] at h,
rw [mem_compl_iff, imp_not_... | lemma | frechet_urysohn_space.of_seq_tendsto_imp_tendsto | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"continuous_at",
"eq_true_iff",
"frechet_urysohn_space",
"imp_not_comm",
"interior_compl",
"mem_interior_iff_mem_nhds",
"nhds_true",
"not_forall",
"not_not",
"subset_seq_closure"
] | An alternative construction for `frechet_urysohn_space`: if sequential convergence implies
convergence, then the space is a Fréchet-Urysohn space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
topological_space.first_countable_topology.frechet_urysohn_space
[first_countable_topology X] : frechet_urysohn_space X | frechet_urysohn_space.of_seq_tendsto_imp_tendsto $ λ f a, tendsto_iff_seq_tendsto.2 | instance | topological_space.first_countable_topology.frechet_urysohn_space | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"frechet_urysohn_space",
"frechet_urysohn_space.of_seq_tendsto_imp_tendsto"
] | Every first-countable space is a Fréchet-Urysohn space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sequential_space (X : Type*) [topological_space X] : Prop | (is_closed_of_seq : ∀ s : set X, is_seq_closed s → is_closed s) | class | sequential_space | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"is_closed",
"is_seq_closed",
"topological_space"
] | A topological space is said to be a *sequential space* if any sequentially closed set in this
space is closed. This condition is weaker than being a Fréchet-Urysohn space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
frechet_urysohn_space.to_sequential_space [frechet_urysohn_space X] :
sequential_space X | ⟨λ s hs, by rw [← closure_eq_iff_is_closed, ← seq_closure_eq_closure, hs.seq_closure_eq]⟩ | instance | frechet_urysohn_space.to_sequential_space | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"closure_eq_iff_is_closed",
"frechet_urysohn_space",
"seq_closure_eq_closure",
"sequential_space"
] | Every Fréchet-Urysohn space is a sequential space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_seq_closed.is_closed [sequential_space X] {s : set X} (hs : is_seq_closed s) :
is_closed s | sequential_space.is_closed_of_seq s hs | lemma | is_seq_closed.is_closed | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"is_closed",
"is_seq_closed",
"sequential_space"
] | In a sequential space, a sequentially closed set is closed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_seq_closed_iff_is_closed [sequential_space X] {M : set X} :
is_seq_closed M ↔ is_closed M | ⟨is_seq_closed.is_closed, is_closed.is_seq_closed⟩ | lemma | is_seq_closed_iff_is_closed | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"is_closed",
"is_seq_closed",
"sequential_space"
] | In a sequential space, a set is closed iff it's sequentially closed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seq_continuous (f : X → Y) : Prop | ∀ ⦃x : ℕ → X⦄ ⦃p : X⦄, tendsto x at_top (𝓝 p) → tendsto (f ∘ x) at_top (𝓝 (f p)) | def | seq_continuous | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [] | A function between topological spaces is sequentially continuous if it commutes with limit of
convergent sequences. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_seq_closed.preimage {f : X → Y} {s : set Y} (hs : is_seq_closed s)
(hf : seq_continuous f) :
is_seq_closed (f ⁻¹' s) | λ x p hx hp, hs hx (hf hp) | lemma | is_seq_closed.preimage | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"is_seq_closed",
"seq_continuous"
] | The preimage of a sequentially closed set under a sequentially continuous map is sequentially
closed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous.seq_continuous {f : X → Y} (hf : continuous f) :
seq_continuous f | λ x p hx, (hf.tendsto p).comp hx | lemma | continuous.seq_continuous | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"continuous",
"seq_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seq_continuous.continuous [sequential_space X] {f : X → Y} (hf : seq_continuous f) :
continuous f | continuous_iff_is_closed.mpr $ λ s hs, (hs.is_seq_closed.preimage hf).is_closed | lemma | seq_continuous.continuous | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"continuous",
"is_closed",
"seq_continuous",
"sequential_space"
] | A sequentially continuous function defined on a sequential space is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_iff_seq_continuous [sequential_space X] {f : X → Y} :
continuous f ↔ seq_continuous f | ⟨continuous.seq_continuous, seq_continuous.continuous⟩ | lemma | continuous_iff_seq_continuous | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"continuous",
"seq_continuous",
"sequential_space"
] | If the domain of a function is a sequential space, then continuity of this function is
equivalent to its sequential continuity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_map.sequential_space [sequential_space X] {f : X → Y} (hf : quotient_map f) :
sequential_space Y | ⟨λ s hs, hf.is_closed_preimage.mp $ (hs.preimage $ hf.continuous.seq_continuous).is_closed⟩ | lemma | quotient_map.sequential_space | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"quotient_map",
"sequential_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_seq_compact (s : set X) | ∀ ⦃x : ℕ → X⦄, (∀ n, x n ∈ s) → ∃ (a ∈ s) (φ : ℕ → ℕ), strict_mono φ ∧ tendsto (x ∘ φ) at_top (𝓝 a) | def | is_seq_compact | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"strict_mono"
] | A set `s` is sequentially compact if every sequence taking values in `s` has a
converging subsequence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
seq_compact_space (X : Type*) [topological_space X] : Prop | (seq_compact_univ : is_seq_compact (univ : set X)) | class | seq_compact_space | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"is_seq_compact",
"topological_space"
] | A space `X` is sequentially compact if every sequence in `X` has a
converging subsequence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_seq_compact.subseq_of_frequently_in {s : set X} (hs : is_seq_compact s) {x : ℕ → X}
(hx : ∃ᶠ n in at_top, x n ∈ s) :
∃ (a ∈ s) (φ : ℕ → ℕ), strict_mono φ ∧ tendsto (x ∘ φ) at_top (𝓝 a) | let ⟨ψ, hψ, huψ⟩ := extraction_of_frequently_at_top hx, ⟨a, a_in, φ, hφ, h⟩ := hs huψ in
⟨a, a_in, ψ ∘ φ, hψ.comp hφ, h⟩ | lemma | is_seq_compact.subseq_of_frequently_in | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"is_seq_compact",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seq_compact_space.tendsto_subseq [seq_compact_space X] (x : ℕ → X) :
∃ a (φ : ℕ → ℕ), strict_mono φ ∧ tendsto (x ∘ φ) at_top (𝓝 a) | let ⟨a, _, φ, mono, h⟩ := seq_compact_univ (λ n, mem_univ (x n)) in ⟨a, φ, mono, h⟩ | lemma | seq_compact_space.tendsto_subseq | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"seq_compact_space",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.is_seq_compact {s : set X} (hs : is_compact s) : is_seq_compact s | λ x x_in, let ⟨a, a_in, ha⟩ := hs (tendsto_principal.mpr (eventually_of_forall x_in))
in ⟨a, a_in, tendsto_subseq ha⟩ | lemma | is_compact.is_seq_compact | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"is_compact",
"is_seq_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.tendsto_subseq' {s : set X} {x : ℕ → X} (hs : is_compact s)
(hx : ∃ᶠ n in at_top, x n ∈ s) :
∃ (a ∈ s) (φ : ℕ → ℕ), strict_mono φ ∧ tendsto (x ∘ φ) at_top (𝓝 a) | hs.is_seq_compact.subseq_of_frequently_in hx | lemma | is_compact.tendsto_subseq' | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"is_compact",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.tendsto_subseq {s : set X} {x : ℕ → X} (hs : is_compact s) (hx : ∀ n, x n ∈ s) :
∃ (a ∈ s) (φ : ℕ → ℕ), strict_mono φ ∧ tendsto (x ∘ φ) at_top (𝓝 a) | hs.is_seq_compact hx | lemma | is_compact.tendsto_subseq | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"is_compact",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
first_countable_topology.seq_compact_of_compact [compact_space X] : seq_compact_space X | ⟨is_compact_univ.is_seq_compact⟩ | instance | first_countable_topology.seq_compact_of_compact | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"compact_space",
"seq_compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compact_space.tendsto_subseq [compact_space X] (x : ℕ → X) :
∃ a (φ : ℕ → ℕ), strict_mono φ ∧ tendsto (x ∘ φ) at_top (𝓝 a) | seq_compact_space.tendsto_subseq x | lemma | compact_space.tendsto_subseq | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"compact_space",
"seq_compact_space.tendsto_subseq",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_seq_compact.exists_tendsto_of_frequently_mem (hs : is_seq_compact s) {u : ℕ → X}
(hu : ∃ᶠ n in at_top, u n ∈ s) (huc : cauchy_seq u) :
∃ x ∈ s, tendsto u at_top (𝓝 x) | let ⟨x, hxs, φ, φ_mono, hx⟩ := hs.subseq_of_frequently_in hu
in ⟨x, hxs, tendsto_nhds_of_cauchy_seq_of_subseq huc φ_mono.tendsto_at_top hx⟩ | lemma | is_seq_compact.exists_tendsto_of_frequently_mem | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"cauchy_seq",
"is_seq_compact",
"tendsto_nhds_of_cauchy_seq_of_subseq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_seq_compact.exists_tendsto (hs : is_seq_compact s) {u : ℕ → X} (hu : ∀ n, u n ∈ s)
(huc : cauchy_seq u) : ∃ x ∈ s, tendsto u at_top (𝓝 x) | hs.exists_tendsto_of_frequently_mem (frequently_of_forall hu) huc | lemma | is_seq_compact.exists_tendsto | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"cauchy_seq",
"is_seq_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_seq_compact.totally_bounded (h : is_seq_compact s) : totally_bounded s | begin
intros V V_in,
unfold is_seq_compact at h,
contrapose! h,
obtain ⟨u, u_in, hu⟩ : ∃ u : ℕ → X, (∀ n, u n ∈ s) ∧ ∀ n m, m < n → u m ∉ ball (u n) V,
{ simp only [not_subset, mem_Union₂, not_exists, exists_prop] at h,
simpa only [forall_and_distrib, ball_image_iff, not_and] using seq_of_forall_finite_ex... | lemma | is_seq_compact.totally_bounded | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"exists_prop",
"forall_and_distrib",
"is_seq_compact",
"le_rfl",
"lt_add_one",
"not_and",
"not_exists",
"totally_bounded"
] | A sequentially compact set in a uniform space is totally bounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_seq_compact.is_complete (hs : is_seq_compact s) : is_complete s | begin
intros l hl hls,
haveI := hl.1,
rcases exists_antitone_basis (𝓤 X) with ⟨V, hV⟩,
choose W hW hWV using λ n, comp_mem_uniformity_sets (hV.mem n),
have hWV' : ∀ n, W n ⊆ V n, from λ n ⟨x, y⟩ hx, @hWV n (x, y) ⟨x, refl_mem_uniformity $ hW _, hx⟩,
obtain ⟨t, ht_anti, htl, htW, hts⟩ : ∃ t : ℕ → set X, ant... | lemma | is_seq_compact.is_complete | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"antitone",
"cauchy_seq",
"comp_mem_uniformity_sets",
"filter.nonempty_of_mem",
"is_complete",
"is_seq_compact",
"le_rfl",
"nhds_basis_uniformity'",
"refl_mem_uniformity"
] | A sequentially compact set in a uniform set with countably generated uniformity filter
is complete. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_seq_compact.is_compact (hs : is_seq_compact s) : is_compact s | is_compact_iff_totally_bounded_is_complete.2 ⟨hs.totally_bounded, hs.is_complete⟩ | lemma | is_seq_compact.is_compact | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"is_compact",
"is_seq_compact"
] | If `𝓤 β` is countably generated, then any sequentially compact set is compact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_space.is_compact_iff_is_seq_compact : is_compact s ↔ is_seq_compact s | ⟨λ H, H.is_seq_compact, λ H, H.is_compact⟩ | lemma | uniform_space.is_compact_iff_is_seq_compact | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"is_compact",
"is_seq_compact"
] | A version of Bolzano-Weistrass: in a uniform space with countably generated uniformity filter
(e.g., in a metric space), a set is compact if and only if it is sequentially compact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_space.compact_space_iff_seq_compact_space : compact_space X ↔ seq_compact_space X | by simp only [← is_compact_univ_iff, seq_compact_space_iff,
uniform_space.is_compact_iff_is_seq_compact] | lemma | uniform_space.compact_space_iff_seq_compact_space | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"compact_space",
"is_compact_univ_iff",
"seq_compact_space",
"uniform_space.is_compact_iff_is_seq_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seq_compact.lebesgue_number_lemma_of_metric {ι : Sort*} {c : ι → set X}
{s : set X} (hs : is_seq_compact s) (hc₁ : ∀ i, is_open (c i)) (hc₂ : s ⊆ ⋃ i, c i) :
∃ δ > 0, ∀ a ∈ s, ∃ i, ball a δ ⊆ c i | lebesgue_number_lemma_of_metric hs.is_compact hc₁ hc₂ | lemma | seq_compact.lebesgue_number_lemma_of_metric | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"is_open",
"is_seq_compact",
"lebesgue_number_lemma_of_metric"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_subseq_of_frequently_bounded (hs : bounded s)
{x : ℕ → X} (hx : ∃ᶠ n in at_top, x n ∈ s) :
∃ a ∈ closure s, ∃ φ : ℕ → ℕ, strict_mono φ ∧ tendsto (x ∘ φ) at_top (𝓝 a) | have hcs : is_seq_compact (closure s), from hs.is_compact_closure.is_seq_compact,
have hu' : ∃ᶠ n in at_top, x n ∈ closure s, from hx.mono (λ n hn, subset_closure hn),
hcs.subseq_of_frequently_in hu' | lemma | tendsto_subseq_of_frequently_bounded | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"closure",
"is_seq_compact",
"strict_mono",
"subset_closure"
] | A version of **Bolzano-Weistrass**: in a proper metric space (eg. $ℝ^n$),
every bounded sequence has a converging subsequence. This version assumes only
that the sequence is frequently in some bounded set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_subseq_of_bounded (hs : bounded s) {x : ℕ → X} (hx : ∀ n, x n ∈ s) :
∃ a ∈ closure s, ∃ φ : ℕ → ℕ, strict_mono φ ∧ tendsto (x ∘ φ) at_top (𝓝 a) | tendsto_subseq_of_frequently_bounded hs $ frequently_of_forall hx | lemma | tendsto_subseq_of_bounded | topology | src/topology/sequences.lean | [
"topology.subset_properties",
"topology.metric_space.basic"
] | [
"closure",
"strict_mono",
"tendsto_subseq_of_frequently_bounded"
] | A version of Bolzano-Weistrass: in a proper metric space (eg. $ℝ^n$),
every bounded sequence has a converging subsequence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
partial_refinement (u : ι → set X) (s : set X) | (to_fun : ι → set X)
(carrier : set ι)
(is_open' : ∀ i, is_open (to_fun i))
(subset_Union' : s ⊆ ⋃ i, to_fun i)
(closure_subset' : ∀ i ∈ carrier, closure (to_fun i) ⊆ (u i))
(apply_eq' : ∀ i ∉ carrier, to_fun i = u i) | structure | shrinking_lemma.partial_refinement | topology | src/topology/shrinking_lemma.lean | [
"topology.separation"
] | [
"closure",
"is_open"
] | Auxiliary definition for the proof of `shrinking_lemma`. A partial refinement of a covering
`⋃ i, u i` of a set `s` is a map `v : ι → set X` and a set `carrier : set ι` such that
* `s ⊆ ⋃ i, v i`;
* all `v i` are open;
* if `i ∈ carrier v`, then `closure (v i) ⊆ u i`;
* if `i ∉ carrier`, then `v i = u i`.
This type i... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subset_Union (v : partial_refinement u s) : s ⊆ ⋃ i, v i | v.subset_Union' | lemma | shrinking_lemma.partial_refinement.subset_Union | topology | src/topology/shrinking_lemma.lean | [
"topology.separation"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_subset (v : partial_refinement u s) {i : ι} (hi : i ∈ v.carrier) :
closure (v i) ⊆ (u i) | v.closure_subset' i hi | lemma | shrinking_lemma.partial_refinement.closure_subset | topology | src/topology/shrinking_lemma.lean | [
"topology.separation"
] | [
"closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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