File: Mathlib/Topology/Compactness/LocallyCompact.lean

full_name: instWeaklyLocallyCompactSpaceProd
code: instance [WeaklyLocallyCompactSpace X] [WeaklyLocallyCompactSpace Y] :
    WeaklyLocallyCompactSpace (X × Y) 
goal: WeaklyLocallyCompactSpace (X × Y)

full_name: instWeaklyLocallyCompactSpaceForallOfFinite
code: instance {ι : Type*} [Finite ι] {X : ι → Type*} [(i : ι) → TopologicalSpace (X i)]
    [(i : ι) → WeaklyLocallyCompactSpace (X i)] :
    WeaklyLocallyCompactSpace ((i : ι) → X i) 
goal: WeaklyLocallyCompactSpace ((i : ι) → X i)

full_name: instWeaklyLocallyCompactSpaceOfCompactSpace
code: instance (priority := 100) [CompactSpace X] : WeaklyLocallyCompactSpace X 
goal: WeaklyLocallyCompactSpace X

full_name: exists_compact_superset
code: /-- In a weakly locally compact space,
every compact set is contained in the interior of a compact set. -/
theorem exists_compact_superset [WeaklyLocallyCompactSpace X] {K : Set X} (hK : IsCompact K) :
    ∃ K', IsCompact K' ∧ K ⊆ interior K' 
goal: ∃ K', IsCompact K' ∧ K ⊆ interior K'

full_name: disjoint_nhds_cocompact
code: /-- In a weakly locally compact space,
the filters `𝓝 x` and `cocompact X` are disjoint for all `X`. -/
theorem disjoint_nhds_cocompact [WeaklyLocallyCompactSpace X] (x : X) :
    Disjoint (𝓝 x) (cocompact X) 
goal: Disjoint (𝓝 x) (Filter.cocompact X)

full_name: compact_basis_nhds
code: theorem compact_basis_nhds [LocallyCompactSpace X] (x : X) :
    (𝓝 x).HasBasis (fun s => s ∈ 𝓝 x ∧ IsCompact s) fun s => s 
goal: (𝓝 x).HasBasis (fun s => s ∈ 𝓝 x ∧ IsCompact s) fun s => s

full_name: local_compact_nhds
code: theorem local_compact_nhds [LocallyCompactSpace X] {x : X} {n : Set X} (h : n ∈ 𝓝 x) :
    ∃ s ∈ 𝓝 x, s ⊆ n ∧ IsCompact s 
goal: ∃ s ∈ 𝓝 x, s ⊆ n ∧ IsCompact s

full_name: LocallyCompactSpace.of_hasBasis
code: theorem LocallyCompactSpace.of_hasBasis {ι : X → Type*} {p : ∀ x, ι x → Prop}
    {s : ∀ x, ι x → Set X} (h : ∀ x, (𝓝 x).HasBasis (p x) (s x))
    (hc : ∀ x i, p x i → IsCompact (s x i)) : LocallyCompactSpace X 
goal: LocallyCompactSpace X

full_name: Prod.locallyCompactSpace
code: instance Prod.locallyCompactSpace (X : Type*) (Y : Type*) [TopologicalSpace X]
    [TopologicalSpace Y] [LocallyCompactSpace X] [LocallyCompactSpace Y] :
    LocallyCompactSpace (X × Y) 
goal: LocallyCompactSpace (X × Y)

full_name: Pi.locallyCompactSpace_of_finite
code: /-- In general it suffices that all but finitely many of the spaces are compact,
  but that's not straightforward to state and use. -/
instance Pi.locallyCompactSpace_of_finite [Finite ι] : LocallyCompactSpace (∀ i, X i) 
goal: LocallyCompactSpace ((i : ι) → X i)

full_name: Pi.locallyCompactSpace
code: /-- For spaces that are not Hausdorff. -/
instance Pi.locallyCompactSpace [∀ i, CompactSpace (X i)] : LocallyCompactSpace (∀ i, X i) 
goal: LocallyCompactSpace ((i : ι) → X i)

full_name: Function.locallyCompactSpace_of_finite
code: instance Function.locallyCompactSpace_of_finite [Finite ι] [LocallyCompactSpace Y] :
    LocallyCompactSpace (ι → Y) 
goal: LocallyCompactSpace (ι → Y)

full_name: Function.locallyCompactSpace
code: instance Function.locallyCompactSpace [LocallyCompactSpace Y] [CompactSpace Y] :
    LocallyCompactSpace (ι → Y) 
goal: LocallyCompactSpace (ι → Y)

full_name: instLocallyCompactPairOfLocallyCompactSpace
code: instance (priority := 900) [LocallyCompactSpace X] : LocallyCompactPair X Y 
goal: LocallyCompactPair X Y

full_name: instWeaklyLocallyCompactSpaceOfLocallyCompactSpace
code: instance (priority := 100) [LocallyCompactSpace X] : WeaklyLocallyCompactSpace X 
goal: WeaklyLocallyCompactSpace X

full_name: exists_compact_subset
code: /-- 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_compact_subset [LocallyCompactSpace X] {x : X} {U : Set X} (hU : IsOpen U)
    (hx : x ∈ U) : ∃ K : Set X, IsCompact K ∧ x ∈ interior K ∧ K ⊆ U 
goal: ∃ K, IsCompact K ∧ x ∈ interior K ∧ K ⊆ U

full_name: exists_mem_nhdsSet_isCompact_mapsTo
code: /-- 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`. -/
lemma exists_mem_nhdsSet_isCompact_mapsTo [LocallyCompactPair X Y] {f : X → Y} {K : Set X}
    {U : Set Y} (hf : Continuous f) (hK : IsCompact K) (hU : IsOpen U) (hKU : MapsTo f K U) :
    ∃ L ∈ 𝓝ˢ K, IsCompact L ∧ MapsTo f L U 
goal: ∃ L ∈ 𝓝ˢ K, IsCompact L ∧ Set.MapsTo f L U

full_name: exists_compact_between
code: /-- 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 exists_compact_between [LocallyCompactSpace X] {K U : Set X} (hK : IsCompact K)
    (hU : IsOpen U) (h_KU : K ⊆ U) : ∃ L, IsCompact L ∧ K ⊆ interior L ∧ L ⊆ U 
goal: ∃ L, IsCompact L ∧ K ⊆ interior L ∧ L ⊆ U

full_name: ClosedEmbedding.locallyCompactSpace
code: protected theorem ClosedEmbedding.locallyCompactSpace [LocallyCompactSpace Y] {f : X → Y}
    (hf : ClosedEmbedding f) : LocallyCompactSpace X 
goal: LocallyCompactSpace X

full_name: IsClosed.locallyCompactSpace
code: protected theorem IsClosed.locallyCompactSpace [LocallyCompactSpace X] {s : Set X}
    (hs : IsClosed s) : LocallyCompactSpace s 
goal: LocallyCompactSpace ↑s

full_name: OpenEmbedding.locallyCompactSpace
code: protected theorem OpenEmbedding.locallyCompactSpace [LocallyCompactSpace Y] {f : X → Y}
    (hf : OpenEmbedding f) : LocallyCompactSpace X 
goal: LocallyCompactSpace X

full_name: IsOpen.locallyCompactSpace
code: protected theorem IsOpen.locallyCompactSpace [LocallyCompactSpace X] {s : Set X} (hs : IsOpen s) :
    LocallyCompactSpace s 
goal: LocallyCompactSpace ↑s