Locally compact spaces

We define the following classes of topological spaces:

• WeaklyLocallyCompactSpace: every point x has a compact neighborhood.
• LocallyCompactSpace: for every point x, every open neighborhood of x contains a compact neighborhood of x. The definition is formulated in terms of the neighborhood filter.

instance instWeaklyLocallyCompactSpaceProd {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [WeaklyLocallyCompactSpace X] [WeaklyLocallyCompactSpace Y] : WeaklyLocallyCompactSpace (X × Y)

Equations

• ⋯ = ⋯

instance instWeaklyLocallyCompactSpaceForallOfFinite {ι : Type u_4} [Finite ι] {X : ι → Type u_5} [(i : ι) → TopologicalSpace (X i)] [∀ (i : ι), WeaklyLocallyCompactSpace (X i)] : WeaklyLocallyCompactSpace ((i : ι) → X i)

Equations

• ⋯ = ⋯

instance instWeaklyLocallyCompactSpaceOfCompactSpace {X : Type u_1} [TopologicalSpace X] [CompactSpace X] : WeaklyLocallyCompactSpace X

Equations

• ⋯ = ⋯

theorem exists_compact_superset {X : Type u_1} [TopologicalSpace X] [WeaklyLocallyCompactSpace X] {K : Set X} (hK : IsCompact K) : ∃ (K' : Set X), IsCompact K' ∧ K ⊆ interior K'

In a weakly locally compact space, every compact set is contained in the interior of a compact set.

theorem disjoint_nhds_cocompact {X : Type u_1} [TopologicalSpace X] [WeaklyLocallyCompactSpace X] (x : X) : Disjoint (nhds x) (Filter.cocompact X)

In a weakly locally compact space, the filters 𝓝 x and cocompact X are disjoint for all X.

theorem compact_basis_nhds {X : Type u_1} [TopologicalSpace X] [LocallyCompactSpace X] (x : X) : (nhds x).HasBasis (fun (s : Set X) => s ∈ nhds x ∧ IsCompact s) fun (s : Set X) => s

theorem local_compact_nhds {X : Type u_1} [TopologicalSpace X] [LocallyCompactSpace X] {x : X} {n : Set X} (h : n ∈ nhds x) : ∃ s ∈ nhds x, s ⊆ n ∧ IsCompact s

theorem LocallyCompactSpace.of_hasBasis {X : Type u_1} [TopologicalSpace X] {ι : X → Type u_4} {p : (x : X) → ι x → Prop} {s : (x : X) → ι x → Set X} (h : ∀ (x : X), (nhds x).HasBasis (p x) (s x)) (hc : ∀ (x : X) (i : ι x), p x i → IsCompact (s x i)) : LocallyCompactSpace X

@[deprecated LocallyCompactSpace.of_hasBasis]

theorem locallyCompactSpace_of_hasBasis {X : Type u_1} [TopologicalSpace X] {ι : X → Type u_4} {p : (x : X) → ι x → Prop} {s : (x : X) → ι x → Set X} (h : ∀ (x : X), (nhds x).HasBasis (p x) (s x)) (hc : ∀ (x : X) (i : ι x), p x i → IsCompact (s x i)) : LocallyCompactSpace X

Alias of LocallyCompactSpace.of_hasBasis.

instance Prod.locallyCompactSpace (X : Type u_4) (Y : Type u_5) [TopologicalSpace X] [TopologicalSpace Y] [LocallyCompactSpace X] [LocallyCompactSpace Y] : LocallyCompactSpace (X × Y)

Equations

• ⋯ = ⋯

instance Pi.locallyCompactSpace_of_finite {ι : Type u_3} {X : ι → Type u_4} [(i : ι) → TopologicalSpace (X i)] [∀ (i : ι), LocallyCompactSpace (X i)] [Finite ι] : LocallyCompactSpace ((i : ι) → X i)

In general it suffices that all but finitely many of the spaces are compact, but that's not straightforward to state and use.

Equations

• ⋯ = ⋯

instance Pi.locallyCompactSpace {ι : Type u_3} {X : ι → Type u_4} [(i : ι) → TopologicalSpace (X i)] [∀ (i : ι), LocallyCompactSpace (X i)] [∀ (i : ι), CompactSpace (X i)] : LocallyCompactSpace ((i : ι) → X i)

For spaces that are not Hausdorff.

Equations

• ⋯ = ⋯

instance Function.locallyCompactSpace_of_finite {Y : Type u_2} {ι : Type u_3} [TopologicalSpace Y] [Finite ι] [LocallyCompactSpace Y] : LocallyCompactSpace (ι → Y)

Equations

• ⋯ = ⋯

instance Function.locallyCompactSpace {Y : Type u_2} {ι : Type u_3} [TopologicalSpace Y] [LocallyCompactSpace Y] [CompactSpace Y] : LocallyCompactSpace (ι → Y)

Equations

• ⋯ = ⋯

instance instLocallyCompactPairOfLocallyCompactSpace {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [LocallyCompactSpace X] : LocallyCompactPair X Y

Equations

• ⋯ = ⋯

instance instWeaklyLocallyCompactSpaceOfLocallyCompactSpace {X : Type u_1} [TopologicalSpace X] [LocallyCompactSpace X] : WeaklyLocallyCompactSpace X

Equations

• ⋯ = ⋯

theorem exists_compact_subset {X : Type u_1} [TopologicalSpace X] [LocallyCompactSpace X] {x : X} {U : Set X} (hU : IsOpen U) (hx : x ∈ U) : ∃ (K : Set X), IsCompact K ∧ x ∈ interior K ∧ K ⊆ U

A reformulation of the definition of locally compact space: In a locally compact space, every open set containing x has a compact subset containing x in its interior.

theorem exists_mem_nhdsSet_isCompact_mapsTo {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [LocallyCompactPair X Y] {f : X → Y} {K : Set X} {U : Set Y} (hf : Continuous f) (hK : IsCompact K) (hU : IsOpen U) (hKU : Set.MapsTo f K U) : ∃ L ∈ nhdsSet K, IsCompact L ∧ Set.MapsTo f L U

If f : X → Y is a continuous map in a locally compact pair of topological spaces, K : set X is a compact set, and U is an open neighbourhood of f '' K, then there exists a compact neighbourhood L of K such that f maps L to U.

This is a generalization of exists_mem_nhds_isCompact_mapsTo.

theorem exists_compact_between {X : Type u_1} [TopologicalSpace X] [LocallyCompactSpace X] {K : Set X} {U : Set X} (hK : IsCompact K) (hU : IsOpen U) (h_KU : K ⊆ U) : ∃ (L : Set X), IsCompact L ∧ K ⊆ interior L ∧ L ⊆ U

In a locally compact space, for every containment K ⊆ U of a compact set K in an open set U, there is a compact neighborhood L such that K ⊆ L ⊆ U: equivalently, there is a compact L such that K ⊆ interior L and L ⊆ U. See also exists_compact_closed_between, in which one guarantees additionally that L is closed if the space is regular.

theorem ClosedEmbedding.locallyCompactSpace {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [LocallyCompactSpace Y] {f : X → Y} (hf : ClosedEmbedding f) : LocallyCompactSpace X

theorem IsClosed.locallyCompactSpace {X : Type u_1} [TopologicalSpace X] [LocallyCompactSpace X] {s : Set X} (hs : IsClosed s) : LocallyCompactSpace ↑s

theorem OpenEmbedding.locallyCompactSpace {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [LocallyCompactSpace Y] {f : X → Y} (hf : OpenEmbedding f) : LocallyCompactSpace X

theorem IsOpen.locallyCompactSpace {X : Type u_1} [TopologicalSpace X] [LocallyCompactSpace X] {s : Set X} (hs : IsOpen s) : LocallyCompactSpace ↑s