fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
inCompact : Bornology X where
cobounded := Filter.cocompact X
le_cofinite := Filter.cocompact_le_cofinite | def | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | inCompact | Sets that are contained in a compact set form a bornology. Its `cobounded` filter is
`Filter.cocompact`. See also `Bornology.relativelyCompact` the bornology of sets with compact
closure. |
inCompact.isBounded_iff : @IsBounded _ (inCompact X) s ↔ ∃ t, IsCompact t ∧ s ⊆ t := by
change sᶜ ∈ Filter.cocompact X ↔ _
rw [Filter.mem_cocompact]
simp | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | inCompact.isBounded_iff | null |
IsCompact.nhdsSet_prod_eq {t : Set Y} (hs : IsCompact s) (ht : IsCompact t) :
𝓝ˢ (s ×ˢ t) = 𝓝ˢ s ×ˢ 𝓝ˢ t := by
simp_rw [hs.nhdsSet_prod_eq_biSup, ht.prod_nhdsSet_eq_biSup, nhdsSet, sSup_image, biSup_prod,
nhds_prod_eq] | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.nhdsSet_prod_eq | If `s` and `t` are compact sets, then the set neighborhoods filter of `s ×ˢ t`
is the product of set neighborhoods filters for `s` and `t`.
For general sets, only the `≤` inequality holds, see `nhdsSet_prod_le`. |
nhdsSet_prod_le_of_disjoint_cocompact {f : Filter Y} (hs : IsCompact s)
(hf : Disjoint f (Filter.cocompact Y)) :
𝓝ˢ s ×ˢ f ≤ 𝓝ˢ (s ×ˢ Set.univ) := by
obtain ⟨K, hKf, hK⟩ := (disjoint_cocompact_right f).mp hf
calc
𝓝ˢ s ×ˢ f
_ ≤ 𝓝ˢ s ×ˢ 𝓟 K := Filter.prod_mono_right _ (Filter.le_principal_... | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | nhdsSet_prod_le_of_disjoint_cocompact | null |
prod_nhdsSet_le_of_disjoint_cocompact {t : Set Y} {f : Filter X} (ht : IsCompact t)
(hf : Disjoint f (Filter.cocompact X)) :
f ×ˢ 𝓝ˢ t ≤ 𝓝ˢ (Set.univ ×ˢ t) := by
obtain ⟨K, hKf, hK⟩ := (disjoint_cocompact_right f).mp hf
calc
f ×ˢ 𝓝ˢ t
_ ≤ (𝓟 K) ×ˢ 𝓝ˢ t := Filter.prod_mono_left _ (Filter.le... | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | prod_nhdsSet_le_of_disjoint_cocompact | null |
nhds_prod_le_of_disjoint_cocompact {f : Filter Y} (x : X)
(hf : Disjoint f (Filter.cocompact Y)) :
𝓝 x ×ˢ f ≤ 𝓝ˢ ({x} ×ˢ Set.univ) := by
simpa using nhdsSet_prod_le_of_disjoint_cocompact isCompact_singleton hf | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | nhds_prod_le_of_disjoint_cocompact | null |
prod_nhds_le_of_disjoint_cocompact {f : Filter X} (y : Y)
(hf : Disjoint f (Filter.cocompact X)) :
f ×ˢ 𝓝 y ≤ 𝓝ˢ (Set.univ ×ˢ {y}) := by
simpa using prod_nhdsSet_le_of_disjoint_cocompact isCompact_singleton hf | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | prod_nhds_le_of_disjoint_cocompact | null |
generalized_tube_lemma (hs : IsCompact s) {t : Set Y} (ht : IsCompact t)
{n : Set (X × Y)} (hn : IsOpen n) (hp : s ×ˢ t ⊆ n) :
∃ (u : Set X) (v : Set Y), IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ u ×ˢ v ⊆ n := by
rw [← hn.mem_nhdsSet, hs.nhdsSet_prod_eq ht,
((hasBasis_nhdsSet _).prod (hasBasis_nhdsSet _)).mem... | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | generalized_tube_lemma | If `s` and `t` are compact sets and `n` is an open neighborhood of `s × t`, then there exist
open neighborhoods `u ⊇ s` and `v ⊇ t` such that `u × v ⊆ n`.
See also `IsCompact.nhdsSet_prod_eq`. |
isCompact_univ_iff : IsCompact (univ : Set X) ↔ CompactSpace X :=
⟨fun h => ⟨h⟩, fun h => h.1⟩ | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | isCompact_univ_iff | null |
isCompact_univ [h : CompactSpace X] : IsCompact (univ : Set X) :=
h.isCompact_univ | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | isCompact_univ | null |
exists_clusterPt_of_compactSpace [CompactSpace X] (f : Filter X) [NeBot f] :
∃ x, ClusterPt x f := by
simpa using isCompact_univ (show f ≤ 𝓟 univ by simp)
nonrec theorem Ultrafilter.le_nhds_lim [CompactSpace X] (F : Ultrafilter X) : ↑F ≤ 𝓝 F.lim := by
rcases isCompact_univ.ultrafilter_le_nhds F (by simp) with... | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | exists_clusterPt_of_compactSpace | null |
CompactSpace.elim_nhds_subcover [CompactSpace X] (U : X → Set X) (hU : ∀ x, U x ∈ 𝓝 x) :
∃ t : Finset X, ⋃ x ∈ t, U x = ⊤ := by
obtain ⟨t, -, s⟩ := IsCompact.elim_nhds_subcover isCompact_univ U fun x _ => hU x
exact ⟨t, top_unique s⟩ | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | CompactSpace.elim_nhds_subcover | null |
compactSpace_of_finite_subfamily_closed
(h : ∀ {ι : Type u} (t : ι → Set X), (∀ i, IsClosed (t i)) → ⋂ i, t i = ∅ →
∃ u : Finset ι, ⋂ i ∈ u, t i = ∅) :
CompactSpace X where
isCompact_univ := isCompact_of_finite_subfamily_closed fun t => by simpa using h t | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | compactSpace_of_finite_subfamily_closed | null |
CompactSpace.iInter_nonempty {ι : Type v} [CompactSpace X] {t : ι → Set X}
(htc : ∀ i, IsClosed (t i))
(hst : ∀ s : Finset ι, (⋂ i ∈ s, t i).Nonempty) :
(⋂ i, t i).Nonempty := by
simpa using IsCompact.inter_iInter_nonempty isCompact_univ t htc (by simpa using hst)
omit [TopologicalSpace X] in | lemma | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | CompactSpace.iInter_nonempty | Given a family of closed sets `t i` in a compact space, if they satisfy the Finite Intersection
Property, then the intersection of all `t i` is nonempty. |
compactSpace_generateFrom [T : TopologicalSpace X] {S : Set (Set X)}
(hTS : T = generateFrom S) (h : ∀ P ⊆ S, ⋃₀ P = univ → ∃ Q ⊆ P, Q.Finite ∧ ⋃₀ Q = univ) :
CompactSpace X := by
rw [← isCompact_univ_iff]
exact isCompact_generateFrom hTS <| by simpa | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | compactSpace_generateFrom | The `CompactSpace` version of **Alexander's subbasis theorem**. If `X` is a topological space with a
subbasis `S`, then `X` is compact if for any open cover of `X` all of whose elements belong to `S`,
there is a finite subcover. |
IsClosed.isCompact [CompactSpace X] (h : IsClosed s) : IsCompact s :=
isCompact_univ.of_isClosed_subset h (subset_univ _) | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsClosed.isCompact | null |
le_nhds_of_unique_clusterPt [CompactSpace X] {l : Filter X} {y : X}
(h : ∀ x, ClusterPt x l → x = y) : l ≤ 𝓝 y :=
isCompact_univ.le_nhds_of_unique_clusterPt univ_mem fun x _ ↦ h x | lemma | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | le_nhds_of_unique_clusterPt | If a filter has a unique cluster point `y` in a compact topological space,
then the filter is less than or equal to `𝓝 y`. |
tendsto_nhds_of_unique_mapClusterPt [CompactSpace X] {Y} {l : Filter Y} {y : X} {f : Y → X}
(h : ∀ x, MapClusterPt x l f → x = y) :
Tendsto f l (𝓝 y) :=
le_nhds_of_unique_clusterPt h | lemma | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | tendsto_nhds_of_unique_mapClusterPt | If `y` is a unique `MapClusterPt` for `f` along `l`
and the codomain of `f` is a compact space,
then `f` tends to `𝓝 y` along `l`. |
noncompact_univ (X : Type*) [TopologicalSpace X] [NoncompactSpace X] :
¬IsCompact (univ : Set X) :=
NoncompactSpace.noncompact_univ | lemma | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | noncompact_univ | null |
IsCompact.ne_univ [NoncompactSpace X] (hs : IsCompact s) : s ≠ univ := fun h =>
noncompact_univ X (h ▸ hs) | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.ne_univ | null |
@[simp]
Filter.cocompact_eq_bot [CompactSpace X] : Filter.cocompact X = ⊥ :=
Filter.hasBasis_cocompact.eq_bot_iff.mpr ⟨Set.univ, isCompact_univ, Set.compl_univ⟩ | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | Filter.cocompact_eq_bot | null |
noncompactSpace_of_neBot (_ : NeBot (Filter.cocompact X)) : NoncompactSpace X :=
⟨fun h' => (Filter.nonempty_of_mem h'.compl_mem_cocompact).ne_empty compl_univ⟩ | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | noncompactSpace_of_neBot | null |
Filter.cocompact_neBot_iff : NeBot (Filter.cocompact X) ↔ NoncompactSpace X :=
⟨noncompactSpace_of_neBot, fun _ => inferInstance⟩ | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | Filter.cocompact_neBot_iff | null |
not_compactSpace_iff : ¬CompactSpace X ↔ NoncompactSpace X :=
⟨fun h₁ => ⟨fun h₂ => h₁ ⟨h₂⟩⟩, fun ⟨h₁⟩ ⟨h₂⟩ => h₁ h₂⟩ | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | not_compactSpace_iff | null |
finite_of_compact_of_discrete [CompactSpace X] [DiscreteTopology X] : Finite X :=
Finite.of_finite_univ <| isCompact_univ.finite_of_discrete | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | finite_of_compact_of_discrete | A compact discrete space is finite. |
Set.Infinite.exists_accPt_cofinite_inf_principal_of_subset_isCompact
{K : Set X} (hs : s.Infinite) (hK : IsCompact K) (hsub : s ⊆ K) :
∃ x ∈ K, AccPt x (cofinite ⊓ 𝓟 s) :=
(@hK _ hs.cofinite_inf_principal_neBot (inf_le_right.trans <| principal_mono.2 hsub)).imp
fun x hx ↦ by rwa [accPt_iff_clusterPt, inf... | lemma | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | Set.Infinite.exists_accPt_cofinite_inf_principal_of_subset_isCompact | null |
Set.Infinite.exists_accPt_of_subset_isCompact {K : Set X} (hs : s.Infinite)
(hK : IsCompact K) (hsub : s ⊆ K) : ∃ x ∈ K, AccPt x (𝓟 s) :=
let ⟨x, hxK, hx⟩ := hs.exists_accPt_cofinite_inf_principal_of_subset_isCompact hK hsub
⟨x, hxK, hx.mono inf_le_right⟩ | lemma | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | Set.Infinite.exists_accPt_of_subset_isCompact | null |
Set.Infinite.exists_accPt_cofinite_inf_principal [CompactSpace X] (hs : s.Infinite) :
∃ x, AccPt x (cofinite ⊓ 𝓟 s) := by
simpa only [mem_univ, true_and]
using hs.exists_accPt_cofinite_inf_principal_of_subset_isCompact isCompact_univ s.subset_univ | lemma | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | Set.Infinite.exists_accPt_cofinite_inf_principal | null |
Set.Infinite.exists_accPt_principal [CompactSpace X] (hs : s.Infinite) : ∃ x, AccPt x (𝓟 s) :=
hs.exists_accPt_cofinite_inf_principal.imp fun _x hx ↦ hx.mono inf_le_right | lemma | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | Set.Infinite.exists_accPt_principal | null |
exists_nhds_ne_neBot (X : Type*) [TopologicalSpace X] [CompactSpace X] [Infinite X] :
∃ z : X, (𝓝[≠] z).NeBot := by
simpa [AccPt] using (@infinite_univ X _).exists_accPt_principal | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | exists_nhds_ne_neBot | null |
finite_cover_nhds_interior [CompactSpace X] {U : X → Set X} (hU : ∀ x, U x ∈ 𝓝 x) :
∃ t : Finset X, ⋃ x ∈ t, interior (U x) = univ :=
let ⟨t, ht⟩ := isCompact_univ.elim_finite_subcover (fun x => interior (U x))
(fun _ => isOpen_interior) fun x _ => mem_iUnion.2 ⟨x, mem_interior_iff_mem_nhds.2 (hU x)⟩
⟨t, u... | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | finite_cover_nhds_interior | null |
finite_cover_nhds [CompactSpace X] {U : X → Set X} (hU : ∀ x, U x ∈ 𝓝 x) :
∃ t : Finset X, ⋃ x ∈ t, U x = univ :=
let ⟨t, ht⟩ := finite_cover_nhds_interior hU
⟨t, univ_subset_iff.1 <| ht.symm.subset.trans <| iUnion₂_mono fun _ _ => interior_subset⟩ | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | finite_cover_nhds | null |
Filter.comap_cocompact_le {f : X → Y} (hf : Continuous f) :
(Filter.cocompact Y).comap f ≤ Filter.cocompact X := by
rw [(Filter.hasBasis_cocompact.comap f).le_basis_iff Filter.hasBasis_cocompact]
intro t ht
refine ⟨f '' t, ht.image hf, ?_⟩
simpa using t.subset_preimage_image f | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | Filter.comap_cocompact_le | The comap of the cocompact filter on `Y` by a continuous function `f : X → Y` is less than or
equal to the cocompact filter on `X`.
This is a reformulation of the fact that images of compact sets are compact. |
disjoint_map_cocompact {g : X → Y} {f : Filter X} (hg : Continuous g)
(hf : Disjoint f (Filter.cocompact X)) : Disjoint (map g f) (Filter.cocompact Y) := by
rw [← Filter.disjoint_comap_iff_map, disjoint_iff_inf_le]
calc
f ⊓ (comap g (cocompact Y))
_ ≤ f ⊓ Filter.cocompact X := inf_le_inf_left f (Filter.... | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | disjoint_map_cocompact | If a filter is disjoint from the cocompact filter, so is its image under any continuous
function. |
isCompact_range [CompactSpace X] {f : X → Y} (hf : Continuous f) : IsCompact (range f) := by
rw [← image_univ]; exact isCompact_univ.image hf | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | isCompact_range | null |
isCompact_diagonal [CompactSpace X] : IsCompact (diagonal X) :=
@range_diag X ▸ isCompact_range (continuous_id.prodMk continuous_id) | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | isCompact_diagonal | null |
isClosedMap_snd_of_compactSpace [CompactSpace X] :
IsClosedMap (Prod.snd : X × Y → Y) := fun s hs => by
rw [← isOpen_compl_iff, isOpen_iff_mem_nhds]
intro y hy
have : univ ×ˢ {y} ⊆ sᶜ := by
exact fun (x, y') ⟨_, rfl⟩ hs => hy ⟨(x, y'), hs, rfl⟩
rcases generalized_tube_lemma isCompact_univ isCompact_sing... | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | isClosedMap_snd_of_compactSpace | If `X` is a compact topological space, then `Prod.snd : X × Y → Y` is a closed map. |
isClosedMap_fst_of_compactSpace [CompactSpace Y] : IsClosedMap (Prod.fst : X × Y → X) :=
isClosedMap_snd_of_compactSpace.comp isClosedMap_swap | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | isClosedMap_fst_of_compactSpace | If `Y` is a compact topological space, then `Prod.fst : X × Y → X` is a closed map. |
exists_subset_nhds_of_compactSpace [CompactSpace X] [Nonempty ι]
{V : ι → Set X} (hV : Directed (· ⊇ ·) V) (hV_closed : ∀ i, IsClosed (V i)) {U : Set X}
(hU : ∀ x ∈ ⋂ i, V i, U ∈ 𝓝 x) : ∃ i, V i ⊆ U :=
exists_subset_nhds_of_isCompact' hV (fun i => (hV_closed i).isCompact) hV_closed hU | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | exists_subset_nhds_of_compactSpace | null |
Topology.IsInducing.isCompact_iff {f : X → Y} (hf : IsInducing f) :
IsCompact s ↔ IsCompact (f '' s) := by
refine ⟨fun hs => hs.image hf.continuous, fun hs F F_ne_bot F_le => ?_⟩
obtain ⟨_, ⟨x, x_in : x ∈ s, rfl⟩, hx : ClusterPt (f x) (map f F)⟩ :=
hs ((map_mono F_le).trans_eq map_principal)
exact ⟨x, x_i... | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | Topology.IsInducing.isCompact_iff | If `f : X → Y` is an inducing map, the image `f '' s` of a set `s` is compact
if and only if `s` is compact. |
Topology.IsEmbedding.isCompact_iff {f : X → Y} (hf : IsEmbedding f) :
IsCompact s ↔ IsCompact (f '' s) := hf.isInducing.isCompact_iff | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | Topology.IsEmbedding.isCompact_iff | If `f : X → Y` is an embedding, the image `f '' s` of a set `s` is compact
if and only if `s` is compact. |
Topology.IsInducing.isCompact_preimage (hf : IsInducing f) (hf' : IsClosed (range f))
{K : Set Y} (hK : IsCompact K) : IsCompact (f ⁻¹' K) := by
replace hK := hK.inter_right hf'
rwa [hf.isCompact_iff, image_preimage_eq_inter_range] | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | Topology.IsInducing.isCompact_preimage | The preimage of a compact set under an inducing map is a compact set. |
Topology.IsInducing.isCompact_preimage_iff {f : X → Y} (hf : IsInducing f) {K : Set Y}
(Kf : K ⊆ range f) : IsCompact (f ⁻¹' K) ↔ IsCompact K := by
rw [hf.isCompact_iff, image_preimage_eq_of_subset Kf] | lemma | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | Topology.IsInducing.isCompact_preimage_iff | null |
Topology.IsInducing.isCompact_preimage' (hf : IsInducing f) {K : Set Y}
(hK : IsCompact K) (Kf : K ⊆ range f) : IsCompact (f ⁻¹' K) :=
(hf.isCompact_preimage_iff Kf).2 hK | lemma | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | Topology.IsInducing.isCompact_preimage' | The preimage of a compact set in the image of an inducing map is compact. |
Topology.IsClosedEmbedding.isCompact_preimage (hf : IsClosedEmbedding f)
{K : Set Y} (hK : IsCompact K) : IsCompact (f ⁻¹' K) :=
hf.isInducing.isCompact_preimage (hf.isClosed_range) hK | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | Topology.IsClosedEmbedding.isCompact_preimage | The preimage of a compact set under a closed embedding is a compact set. |
Topology.IsClosedEmbedding.tendsto_cocompact (hf : IsClosedEmbedding f) :
Tendsto f (Filter.cocompact X) (Filter.cocompact Y) :=
Filter.hasBasis_cocompact.tendsto_right_iff.mpr fun _K hK =>
(hf.isCompact_preimage hK).compl_mem_cocompact | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | Topology.IsClosedEmbedding.tendsto_cocompact | A closed embedding is proper, i.e., inverse images of compact sets are contained in compacts.
Moreover, the preimage of a compact set is compact, see `IsClosedEmbedding.isCompact_preimage`. |
Subtype.isCompact_iff {p : X → Prop} {s : Set { x // p x }} :
IsCompact s ↔ IsCompact ((↑) '' s : Set X) :=
IsEmbedding.subtypeVal.isCompact_iff | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | Subtype.isCompact_iff | Sets of subtype are compact iff the image under a coercion is. |
isCompact_iff_isCompact_univ : IsCompact s ↔ IsCompact (univ : Set s) := by
rw [Subtype.isCompact_iff, image_univ, Subtype.range_coe] | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | isCompact_iff_isCompact_univ | null |
isCompact_iff_compactSpace : IsCompact s ↔ CompactSpace s :=
isCompact_iff_isCompact_univ.trans isCompact_univ_iff | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | isCompact_iff_compactSpace | null |
IsCompact.finite (hs : IsCompact s) (hs' : DiscreteTopology s) : s.Finite :=
finite_coe_iff.mp (@finite_of_compact_of_discrete _ _ (isCompact_iff_compactSpace.mp hs) hs') | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.finite | null |
exists_nhds_ne_inf_principal_neBot (hs : IsCompact s) (hs' : s.Infinite) :
∃ z ∈ s, (𝓝[≠] z ⊓ 𝓟 s).NeBot :=
hs'.exists_accPt_of_subset_isCompact hs Subset.rfl | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | exists_nhds_ne_inf_principal_neBot | null |
protected Topology.IsClosedEmbedding.noncompactSpace [NoncompactSpace X] {f : X → Y}
(hf : IsClosedEmbedding f) : NoncompactSpace Y :=
noncompactSpace_of_neBot hf.tendsto_cocompact.neBot | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | Topology.IsClosedEmbedding.noncompactSpace | null |
protected Topology.IsClosedEmbedding.compactSpace [h : CompactSpace Y] {f : X → Y}
(hf : IsClosedEmbedding f) : CompactSpace X :=
⟨by rw [hf.isInducing.isCompact_iff, image_univ]; exact hf.isClosed_range.isCompact⟩ | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | Topology.IsClosedEmbedding.compactSpace | null |
IsCompact.prod {t : Set Y} (hs : IsCompact s) (ht : IsCompact t) :
IsCompact (s ×ˢ t) := by
rw [isCompact_iff_ultrafilter_le_nhds'] at hs ht ⊢
intro f hfs
obtain ⟨x : X, sx : x ∈ s, hx : map Prod.fst f.1 ≤ 𝓝 x⟩ :=
hs (f.map Prod.fst) (mem_map.2 <| mem_of_superset hfs fun x => And.left)
obtain ⟨y : Y, t... | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.prod | null |
Filter.coprod_cocompact :
(Filter.cocompact X).coprod (Filter.cocompact Y) = Filter.cocompact (X × Y) := by
apply le_antisymm
· exact sup_le (comap_cocompact_le continuous_fst) (comap_cocompact_le continuous_snd)
· refine (hasBasis_cocompact.coprod hasBasis_cocompact).ge_iff.2 fun K hK ↦ ?_
rw [← univ_pro... | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | Filter.coprod_cocompact | Finite topological spaces are compact. -/
instance (priority := 100) Finite.compactSpace [Finite X] : CompactSpace X where
isCompact_univ := finite_univ.isCompact
instance ULift.compactSpace [CompactSpace X] : CompactSpace (ULift.{v} X) :=
IsClosedEmbedding.uliftDown.compactSpace
/-- The product of two compact sp... |
Prod.noncompactSpace_iff :
NoncompactSpace (X × Y) ↔ NoncompactSpace X ∧ Nonempty Y ∨ Nonempty X ∧ NoncompactSpace Y := by
simp [← Filter.cocompact_neBot_iff, ← Filter.coprod_cocompact, Filter.coprod_neBot_iff] | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | Prod.noncompactSpace_iff | null |
isCompact_pi_infinite {s : ∀ i, Set (X i)} :
(∀ i, IsCompact (s i)) → IsCompact { x : ∀ i, X i | ∀ i, x i ∈ s i } := by
simp only [isCompact_iff_ultrafilter_le_nhds, nhds_pi, le_pi, le_principal_iff]
intro h f hfs
have : ∀ i : ι, ∃ x, x ∈ s i ∧ Tendsto (Function.eval i) f (𝓝 x) := by
refine fun i => h i ... | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | isCompact_pi_infinite | **Tychonoff's theorem**: product of compact sets is compact. |
isCompact_univ_pi {s : ∀ i, Set (X i)} (h : ∀ i, IsCompact (s i)) :
IsCompact (pi univ s) := by
convert isCompact_pi_infinite h
simp only [← mem_univ_pi, setOf_mem_eq] | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | isCompact_univ_pi | **Tychonoff's theorem** formulated using `Set.pi`: product of compact sets is compact. |
Pi.compactSpace [∀ i, CompactSpace (X i)] : CompactSpace (∀ i, X i) :=
⟨by rw [← pi_univ univ]; exact isCompact_univ_pi fun i => isCompact_univ⟩ | instance | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | Pi.compactSpace | null |
Function.compactSpace [CompactSpace Y] : CompactSpace (ι → Y) :=
Pi.compactSpace | instance | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | Function.compactSpace | null |
Pi.isCompact_iff_of_isClosed {s : Set (Π i, X i)} (hs : IsClosed s) :
IsCompact s ↔ ∀ i, IsCompact (eval i '' s) := by
constructor <;> intro H
· exact fun i ↦ H.image <| continuous_apply i
· exact IsCompact.of_isClosed_subset (isCompact_univ_pi H) hs (subset_pi_eval_image univ s) | lemma | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | Pi.isCompact_iff_of_isClosed | null |
protected Pi.exists_compact_superset_iff {s : Set (Π i, X i)} :
(∃ K, IsCompact K ∧ s ⊆ K) ↔ ∀ i, ∃ Ki, IsCompact Ki ∧ s ⊆ eval i ⁻¹' Ki := by
constructor
· intro ⟨K, hK, hsK⟩ i
exact ⟨eval i '' K, hK.image <| continuous_apply i, hsK.trans <| K.subset_preimage_image _⟩
· intro H
choose K hK hsK using ... | lemma | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | Pi.exists_compact_superset_iff | null |
Filter.coprodᵢ_cocompact {X : ι → Type*} [∀ d, TopologicalSpace (X d)] :
(Filter.coprodᵢ fun d => Filter.cocompact (X d)) = Filter.cocompact (∀ d, X d) := by
refine le_antisymm (iSup_le fun i => Filter.comap_cocompact_le (continuous_apply i)) ?_
refine compl_surjective.forall.2 fun s H => ?_
simp only [compl_... | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | Filter.coprodᵢ_cocompact | **Tychonoff's theorem** formulated in terms of filters: `Filter.cocompact` on an indexed product
type `Π d, X d` the `Filter.coprodᵢ` of filters `Filter.cocompact` on `X d`. |
Quot.compactSpace {r : X → X → Prop} [CompactSpace X] : CompactSpace (Quot r) :=
⟨by
rw [← range_quot_mk]
exact isCompact_range continuous_quot_mk⟩ | instance | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | Quot.compactSpace | null |
Quotient.compactSpace {s : Setoid X} [CompactSpace X] : CompactSpace (Quotient s) :=
Quot.compactSpace | instance | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | Quotient.compactSpace | null |
IsClosed.exists_minimal_nonempty_closed_subset [CompactSpace X] {S : Set X}
(hS : IsClosed S) (hne : S.Nonempty) :
∃ V : Set X, V ⊆ S ∧ V.Nonempty ∧ IsClosed V ∧
∀ V' : Set X, V' ⊆ V → V'.Nonempty → IsClosed V' → V' = V := by
let opens := { U : Set X | Sᶜ ⊆ U ∧ IsOpen U ∧ Uᶜ.Nonempty }
obtain ⟨U, h⟩ :... | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsClosed.exists_minimal_nonempty_closed_subset | null |
CompactlyCoherentSpace (X : Type*) [TopologicalSpace X] : Prop where
/-- A space is a compactly coherent space if the topology is generated by the compact sets. -/
isCoherentWith : IsCoherentWith (X := X) {K | IsCompact K} | class | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.Compactness.Compact"
] | Mathlib/Topology/Compactness/CompactlyCoherentSpace.lean | CompactlyCoherentSpace | A space is a compactly coherent space if the topology is generated by the compact sets. |
isOpen_iff [CompactlyCoherentSpace X] {A : Set X} :
IsOpen A ↔ ∀ K, IsCompact K → IsOpen (K ↓∩ A) :=
IsCoherentWith.isOpen_iff isCoherentWith | lemma | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.Compactness.Compact"
] | Mathlib/Topology/Compactness/CompactlyCoherentSpace.lean | isOpen_iff | A set `A` in a compactly coherent space is open iff for every compact set `K`, the intersection
`K ∩ A` is open in `K`. |
isClosed_iff [CompactlyCoherentSpace X] (A : Set X) :
IsClosed A ↔ ∀ K, IsCompact K → IsClosed (K ↓∩ A) :=
IsCoherentWith.isClosed_iff isCoherentWith | lemma | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.Compactness.Compact"
] | Mathlib/Topology/Compactness/CompactlyCoherentSpace.lean | isClosed_iff | A set `A` in a compactly coherent space is closed iff for every compact set `K`, the
intersection `K ∩ A` is closed in `K`. |
of_isOpen (h : ∀ (A : Set X), (∀ K, IsCompact K → IsOpen (K ↓∩ A)) → IsOpen A) :
CompactlyCoherentSpace X where
isCoherentWith := {isOpen_of_forall_induced := h} | lemma | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.Compactness.Compact"
] | Mathlib/Topology/Compactness/CompactlyCoherentSpace.lean | of_isOpen | If every set `A` is open if for every compact `K` the intersection `K ∩ A` is open in `K`,
then the space is a compactly coherent space. |
of_isClosed (h : ∀ (A : Set X), (∀ K, IsCompact K → IsClosed (K ↓∩ A)) → IsClosed A) :
CompactlyCoherentSpace X where
isCoherentWith := IsCoherentWith.of_isClosed h | lemma | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.Compactness.Compact"
] | Mathlib/Topology/Compactness/CompactlyCoherentSpace.lean | of_isClosed | If every set `A` is closed if for every compact `K` the intersection `K ∩ A` is closed in `K`,
then the space is a compactly coherent space. |
of_weaklyLocallyCompactSpace [WeaklyLocallyCompactSpace X] : CompactlyCoherentSpace X where
isCoherentWith := IsCoherentWith.of_nhds exists_compact_mem_nhds
@[deprecated (since := "2025-05-30")] alias
_root_.Topology.IsCoherentWith.isCompact_of_weaklyLocallyCompact := of_weaklyLocallyCompactSpace | instance | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.Compactness.Compact"
] | Mathlib/Topology/Compactness/CompactlyCoherentSpace.lean | of_weaklyLocallyCompactSpace | Every weakly locally compact space is a compactly coherent space. |
of_sequentialSpace [SequentialSpace X] : CompactlyCoherentSpace X where
isCoherentWith := IsCoherentWith.of_seq fun _u _x hux ↦ hux.isCompact_insert_range
@[deprecated (since := "2025-05-30")] alias
_root_.Topology.IsCoherentWith.isCompact_of_seq := of_sequentialSpace | instance | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.Compactness.Compact"
] | Mathlib/Topology/Compactness/CompactlyCoherentSpace.lean | of_sequentialSpace | Every sequential space is a compactly coherent space. |
isOpen_iff_forall_compactSpace [CompactlyCoherentSpace X] (s : Set X) :
IsOpen s ↔
∀ (K : Type u) [TopologicalSpace K] [CompactSpace K],
∀ (f : K → X), Continuous f → IsOpen (f ⁻¹' s) := by
refine ⟨fun hs _ _ _ _ hf ↦ hs.preimage hf, fun hs ↦ isOpen_iff |>.mpr ?_⟩
intro K hK
have : CompactSpace K ... | lemma | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.Compactness.Compact"
] | Mathlib/Topology/Compactness/CompactlyCoherentSpace.lean | isOpen_iff_forall_compactSpace | In a compactly coherent space `X`, a set `s` is open iff `f ⁻¹' s` is open for every continuous
map from a compact space. |
of_isOpen_forall_compactSpace (h : ∀ (s : Set X), (∀ (K : Type u) [TopologicalSpace K],
[CompactSpace K] → ∀ (f : K → X), Continuous f → IsOpen (f ⁻¹' s)) → IsOpen s) :
CompactlyCoherentSpace X := by
refine of_isOpen fun A hA ↦ h A fun K _ _ f hf ↦ ?_
specialize hA (range f) (isCompact_range hf)
have :=... | lemma | Topology | [
"Mathlib.Topology.Coherent",
"Mathlib.Topology.Compactness.Compact"
] | Mathlib/Topology/Compactness/CompactlyCoherentSpace.lean | of_isOpen_forall_compactSpace | A topological space `X` is compactly coherent if a set `s` is open when `f ⁻¹' s?` is open for
every continuous map `f : K → X`, where `K` is compact. |
TopologicalSpace.compactlyGenerated (X : Type w) [TopologicalSpace X] : TopologicalSpace X :=
let f : (Σ (i : (S : CompHaus.{u}) × C(S, X)), i.fst) → X := fun ⟨⟨_, i⟩, s⟩ ↦ i s
coinduced f inferInstance | def | Topology | [
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean | TopologicalSpace.compactlyGenerated | The compactly generated topology on a topological space `X`. This is the finest topology
which makes all maps from compact Hausdorff spaces to `X`, which are continuous for the original
topology, continuous.
Note: this definition should be used with an explicit universe parameter `u` for the size of the
compact Hausdo... |
continuous_from_compactlyGenerated [TopologicalSpace X] [t : TopologicalSpace Y] (f : X → Y)
(h : ∀ (S : CompHaus.{u}) (g : C(S, X)), Continuous (f ∘ g)) :
Continuous[compactlyGenerated.{u} X, t] f := by
rw [continuous_coinduced_dom]
continuity | lemma | Topology | [
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean | continuous_from_compactlyGenerated | null |
UCompactlyGeneratedSpace (X : Type v) [t : TopologicalSpace X] : Prop where
/-- The topology of `X` is finer than the compactly generated topology. -/
le_compactlyGenerated : t ≤ compactlyGenerated.{u} X | class | Topology | [
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean | UCompactlyGeneratedSpace | A topological space `X` is compactly generated if its topology is finer than (and thus equal to)
the compactly generated topology, i.e. it is coinduced by the continuous maps from compact
Hausdorff spaces to `X`.
This version includes an explicit universe parameter `u` which should always be specified. It is
intended ... |
eq_compactlyGenerated [t : TopologicalSpace X] [UCompactlyGeneratedSpace.{u} X] :
t = compactlyGenerated.{u} X := by
apply le_antisymm
· exact UCompactlyGeneratedSpace.le_compactlyGenerated
· simp only [compactlyGenerated, ← continuous_iff_coinduced_le, continuous_sigma_iff,
Sigma.forall]
exact fun ... | lemma | Topology | [
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean | eq_compactlyGenerated | null |
uCompactlyGeneratedSpace_of_continuous_maps [t : TopologicalSpace X]
(h : ∀ {Y : Type w} [tY : TopologicalSpace Y] (f : X → Y),
(∀ (S : CompHaus.{u}) (g : C(S, X)), Continuous (f ∘ g)) → Continuous f) :
UCompactlyGeneratedSpace.{u} X where
le_compactlyGenerated := by
suffices Continuous[t, compa... | lemma | Topology | [
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean | uCompactlyGeneratedSpace_of_continuous_maps | Let `f : X → Y`. Suppose that to prove that `f` is continuous, it suffices to show that
for every compact Hausdorff space `K` and every continuous map `g : K → X`, `f ∘ g` is continuous.
Then `X` is compactly generated. |
continuous_from_uCompactlyGeneratedSpace [UCompactlyGeneratedSpace.{u} X] (f : X → Y)
(h : ∀ (S : CompHaus.{u}) (g : C(S, X)), Continuous (f ∘ g)) : Continuous f := by
apply continuous_le_dom UCompactlyGeneratedSpace.le_compactlyGenerated
exact continuous_from_compactlyGenerated f h | lemma | Topology | [
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean | continuous_from_uCompactlyGeneratedSpace | If `X` is compactly generated, to prove that `f : X → Y` is continuous it is enough to show
that for every compact Hausdorff space `K` and every continuous map `g : K → X`,
`f ∘ g` is continuous. |
uCompactlyGeneratedSpace_of_isClosed
(h : ∀ (s : Set X), (∀ (S : CompHaus.{u}) (f : C(S, X)), IsClosed (f ⁻¹' s)) → IsClosed s) :
UCompactlyGeneratedSpace.{u} X :=
uCompactlyGeneratedSpace_of_continuous_maps fun _ h' ↦
continuous_iff_isClosed.2 fun _ hs ↦ h _ fun S g ↦ hs.preimage (h' S g) | theorem | Topology | [
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean | uCompactlyGeneratedSpace_of_isClosed | A topological space `X` is compactly generated if a set `s` is closed when `f ⁻¹' s` is
closed for every continuous map `f : K → X`, where `K` is compact Hausdorff. |
uCompactlyGeneratedSpace_of_isOpen
(h : ∀ (s : Set X), (∀ (S : CompHaus.{u}) (f : C(S, X)), IsOpen (f ⁻¹' s)) → IsOpen s) :
UCompactlyGeneratedSpace.{u} X :=
uCompactlyGeneratedSpace_of_continuous_maps fun _ h' ↦
continuous_def.2 fun _ hs ↦ h _ fun S g ↦ hs.preimage (h' S g) | theorem | Topology | [
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean | uCompactlyGeneratedSpace_of_isOpen | A topological space `X` is compactly generated if a set `s` is open when `f ⁻¹' s` is
open for every continuous map `f : K → X`, where `K` is compact Hausdorff. |
UCompactlyGeneratedSpace.isClosed [UCompactlyGeneratedSpace.{u} X] {s : Set X}
(hs : ∀ (S : CompHaus.{u}) (f : C(S, X)), IsClosed (f ⁻¹' s)) : IsClosed s := by
rw [eq_compactlyGenerated (X := X), TopologicalSpace.compactlyGenerated, isClosed_coinduced,
isClosed_sigma_iff]
exact fun ⟨S, f⟩ ↦ hs S f | theorem | Topology | [
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean | UCompactlyGeneratedSpace.isClosed | In a compactly generated space `X`, a set `s` is closed when `f ⁻¹' s` is
closed for every continuous map `f : K → X`, where `K` is compact Hausdorff. |
UCompactlyGeneratedSpace.isOpen [UCompactlyGeneratedSpace.{u} X] {s : Set X}
(hs : ∀ (S : CompHaus.{u}) (f : C(S, X)), IsOpen (f ⁻¹' s)) : IsOpen s := by
rw [eq_compactlyGenerated (X := X), TopologicalSpace.compactlyGenerated, isOpen_coinduced,
isOpen_sigma_iff]
exact fun ⟨S, f⟩ ↦ hs S f | theorem | Topology | [
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean | UCompactlyGeneratedSpace.isOpen | In a compactly generated space `X`, a set `s` is open when `f ⁻¹' s` is
open for every continuous map `f : K → X`, where `K` is compact Hausdorff. |
uCompactlyGeneratedSpace_of_coinduced
[UCompactlyGeneratedSpace.{u} X] {f : X → Y} (hf : Continuous f) (ht : tY = coinduced f tX) :
UCompactlyGeneratedSpace.{u} Y := by
refine uCompactlyGeneratedSpace_of_isClosed fun s h ↦ ?_
rw [ht, isClosed_coinduced]
exact UCompactlyGeneratedSpace.isClosed fun _ ⟨g, hg... | theorem | Topology | [
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean | uCompactlyGeneratedSpace_of_coinduced | If the topology of `X` is coinduced by a continuous function whose domain is
compactly generated, then so is `X`. |
CompactlyGeneratedSpace (X : Type u) [TopologicalSpace X] : Prop :=
UCompactlyGeneratedSpace.{u} X | abbrev | Topology | [
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean | CompactlyGeneratedSpace | The quotient of a compactly generated space is compactly generated. -/
instance {S : Setoid X} [UCompactlyGeneratedSpace.{u} X] :
UCompactlyGeneratedSpace.{u} (Quotient S) :=
uCompactlyGeneratedSpace_of_coinduced continuous_quotient_mk' rfl
/-- The sum of two compactly generated spaces is compactly generated. -/... |
continuous_from_compactlyGeneratedSpace [CompactlyGeneratedSpace X] (f : X → Y)
(h : ∀ (K : Type u) [TopologicalSpace K], [CompactSpace K] → [T2Space K] →
(∀ g : K → X, Continuous g → Continuous (f ∘ g))) : Continuous f :=
continuous_from_uCompactlyGeneratedSpace f fun K ⟨g, hg⟩ ↦ h K g hg | lemma | Topology | [
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean | continuous_from_compactlyGeneratedSpace | If `X` is compactly generated, to prove that `f : X → Y` is continuous it is enough to show
that for every compact Hausdorff space `K` and every continuous map `g : K → X`,
`f ∘ g` is continuous. |
compactlyGeneratedSpace_of_continuous_maps
(h : ∀ {Y : Type u} [TopologicalSpace Y] (f : X → Y),
(∀ (K : Type u) [TopologicalSpace K], [CompactSpace K] → [T2Space K] →
(∀ g : K → X, Continuous g → Continuous (f ∘ g))) → Continuous f) :
CompactlyGeneratedSpace X :=
uCompactlyGeneratedSpace_of_con... | lemma | Topology | [
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean | compactlyGeneratedSpace_of_continuous_maps | Let `f : X → Y`. Suppose that to prove that `f` is continuous, it suffices to show that
for every compact Hausdorff space `K` and every continuous map `g : K → X`, `f ∘ g` is continuous.
Then `X` is compactly generated. |
compactlyGeneratedSpace_of_isClosed
(h : ∀ (s : Set X), (∀ (K : Type u) [TopologicalSpace K], [CompactSpace K] → [T2Space K] →
∀ (f : K → X), Continuous f → IsClosed (f ⁻¹' s)) → IsClosed s) :
CompactlyGeneratedSpace X :=
uCompactlyGeneratedSpace_of_isClosed fun s h' ↦ h s fun K _ _ _ f hf ↦ h' (CompHau... | theorem | Topology | [
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean | compactlyGeneratedSpace_of_isClosed | A topological space `X` is compactly generated if a set `s` is closed when `f ⁻¹' s` is
closed for every continuous map `f : K → X`, where `K` is compact Hausdorff. |
CompactlyGeneratedSpace.isClosed' [CompactlyGeneratedSpace X] {s : Set X}
(hs : ∀ (K : Type u) [TopologicalSpace K], [CompactSpace K] → [T2Space K] →
∀ (f : K → X), Continuous f → IsClosed (f ⁻¹' s)) : IsClosed s :=
UCompactlyGeneratedSpace.isClosed fun S ⟨f, hf⟩ ↦ hs S f hf | theorem | Topology | [
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean | CompactlyGeneratedSpace.isClosed' | In a compactly generated space `X`, a set `s` is closed when `f ⁻¹' s` is
closed for every continuous map `f : K → X`, where `K` is compact Hausdorff. |
CompactlyGeneratedSpace.isClosed [CompactlyGeneratedSpace X] {s : Set X}
(hs : ∀ ⦃K⦄, IsCompact K → IsClosed (s ∩ K)) : IsClosed s := by
refine isClosed' fun K _ _ _ f hf ↦ ?_
rw [← Set.preimage_inter_range]
exact (hs (isCompact_range hf)).preimage hf | theorem | Topology | [
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean | CompactlyGeneratedSpace.isClosed | In a compactly generated space `X`, a set `s` is closed when `s ∩ K` is
closed for every compact set `K`. |
compactlyGeneratedSpace_of_isOpen
(h : ∀ (s : Set X), (∀ (K : Type u) [TopologicalSpace K], [CompactSpace K] → [T2Space K] →
∀ (f : K → X), Continuous f → IsOpen (f ⁻¹' s)) → IsOpen s) :
CompactlyGeneratedSpace X :=
uCompactlyGeneratedSpace_of_isOpen fun s h' ↦ h s fun K _ _ _ f hf ↦ h' (CompHaus.of K) ... | theorem | Topology | [
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean | compactlyGeneratedSpace_of_isOpen | A topological space `X` is compactly generated if a set `s` is open when `f ⁻¹' s` is
open for every continuous map `f : K → X`, where `K` is compact Hausdorff. |
CompactlyGeneratedSpace.isOpen' [CompactlyGeneratedSpace X] {s : Set X}
(hs : ∀ (K : Type u) [TopologicalSpace K], [CompactSpace K] → [T2Space K] →
∀ (f : K → X), Continuous f → IsOpen (f ⁻¹' s)) : IsOpen s :=
UCompactlyGeneratedSpace.isOpen fun S ⟨f, hf⟩ ↦ hs S f hf | theorem | Topology | [
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean | CompactlyGeneratedSpace.isOpen' | In a compactly generated space `X`, a set `s` is open when `f ⁻¹' s` is
open for every continuous map `f : K → X`, where `K` is compact Hausdorff. |
CompactlyGeneratedSpace.isOpen [CompactlyGeneratedSpace X] {s : Set X}
(hs : ∀ ⦃K⦄, IsCompact K → IsOpen (s ∩ K)) : IsOpen s := by
refine isOpen' fun K _ _ _ f hf ↦ ?_
rw [← Set.preimage_inter_range]
exact (hs (isCompact_range hf)).preimage hf | theorem | Topology | [
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean | CompactlyGeneratedSpace.isOpen | In a compactly generated space `X`, a set `s` is open when `s ∩ K` is
closed for every open set `K`. |
compactlyGeneratedSpace_of_coinduced
{X : Type u} [tX : TopologicalSpace X] {Y : Type u} [tY : TopologicalSpace Y]
[CompactlyGeneratedSpace X] {f : X → Y} (hf : Continuous f) (ht : tY = coinduced f tX) :
CompactlyGeneratedSpace Y := uCompactlyGeneratedSpace_of_coinduced hf ht | theorem | Topology | [
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean | compactlyGeneratedSpace_of_coinduced | If the topology of `X` is coinduced by a continuous function whose domain is
compactly generated, then so is `X`. |
compactlyGeneratedSpace_of_isClosed_of_t2
(h : ∀ s, (∀ (K : Set X), IsCompact K → IsClosed (s ∩ K)) → IsClosed s) :
CompactlyGeneratedSpace X := by
refine compactlyGeneratedSpace_of_isClosed fun s hs ↦ h s fun K hK ↦ ?_
rw [Set.inter_comm, ← Subtype.image_preimage_coe]
apply hK.isClosed.isClosedMap_subtyp... | theorem | Topology | [
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean | compactlyGeneratedSpace_of_isClosed_of_t2 | The sigma type associated to a family of compactly generated spaces is compactly generated. -/
instance {ι : Type u} {X : ι → Type v}
[∀ i, TopologicalSpace (X i)] [∀ i, CompactlyGeneratedSpace (X i)] :
CompactlyGeneratedSpace (Σ i, X i) := by
refine compactlyGeneratedSpace_of_isClosed fun s h ↦ isClosed_sigm... |
compactlyGeneratedSpace_of_isOpen_of_t2
(h : ∀ s, (∀ (K : Set X), IsCompact K → IsOpen (K ↓∩ s)) → IsOpen s) :
CompactlyGeneratedSpace X := by
refine compactlyGeneratedSpace_of_isOpen fun s hs ↦ h s fun K hK ↦ ?_
have : CompactSpace ↑K := isCompact_iff_compactSpace.1 hK
exact hs _ Subtype.val continuous_s... | theorem | Topology | [
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean | compactlyGeneratedSpace_of_isOpen_of_t2 | Let `s ⊆ X`. Suppose that `X` is Hausdorff, and that to prove that `s` is open,
it suffices to show that for every compact set `K ⊆ X`, `s ∩ K` is open in `K`.
Then `X` is compactly generated. |
to_compactlyCoherentSpace [CompactlyGeneratedSpace X] : CompactlyCoherentSpace X :=
CompactlyCoherentSpace.of_isOpen_forall_compactSpace fun _ h ↦ CompactlyGeneratedSpace.isOpen'
fun K _ _ _ f hf ↦ h K f hf | instance | Topology | [
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean | to_compactlyCoherentSpace | A Hausdorff and weakly locally compact space is compactly generated. -/
instance (priority := 100) [WeaklyLocallyCompactSpace X] :
CompactlyGeneratedSpace X := by
refine compactlyGeneratedSpace_of_isClosed_of_t2 fun s h ↦ ?_
rw [isClosed_iff_forall_filter]
intro x ℱ hℱ₁ hℱ₂ hℱ₃
rcases exists_compact_mem_nhd... |
of_compactlyCoherentSpace_of_t2 [T2Space X] [CompactlyCoherentSpace X] :
CompactlyGeneratedSpace X := by
apply compactlyGeneratedSpace_of_isClosed_of_t2
intro s hs
rw [CompactlyCoherentSpace.isClosed_iff]
intro K hK
rw [← Subtype.preimage_coe_inter_self]
exact (hs K hK).preimage_val | instance | Topology | [
"Mathlib.Topology.Category.CompHaus.Basic",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] | Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean | of_compactlyCoherentSpace_of_t2 | A compactly coherent space that is Hausdorff is compactly generated. |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.