Documentation

Mathlib.Topology.Sets.Compacts

Compact sets #

We define a few types of compact sets in a topological space.

Main Definitions #

For a topological space α,

Compact sets #

Equations
  • TopologicalSpace.Compacts.instSetLike = { coe := TopologicalSpace.Compacts.carrier, coe_injective' := }

See Note [custom simps projection].

Equations
Equations
  • =
@[simp]
theorem TopologicalSpace.Compacts.coe_mk {α : Type u_1} [TopologicalSpace α] (s : Set α) (h : IsCompact s) :
{ carrier := s, isCompact' := h } = s
Equations
Equations
  • TopologicalSpace.Compacts.instInfOfT2Space = { inf := fun (s t : TopologicalSpace.Compacts α) => { carrier := s t, isCompact' := } }
Equations
  • TopologicalSpace.Compacts.instTopOfCompactSpace = { top := { carrier := Set.univ, isCompact' := } }
Equations
  • TopologicalSpace.Compacts.instBot = { bot := { carrier := , isCompact' := } }
Equations
Equations
Equations
  • TopologicalSpace.Compacts.instOrderBot = OrderBot.lift SetLike.coe
Equations
  • TopologicalSpace.Compacts.instBoundedOrderOfCompactSpace = BoundedOrder.lift SetLike.coe

The type of compact sets is inhabited, with default element the empty set.

Equations
  • TopologicalSpace.Compacts.instInhabited = { default := }
@[simp]
@[simp]
theorem TopologicalSpace.Compacts.coe_top {α : Type u_1} [TopologicalSpace α] [CompactSpace α] :
= Set.univ
@[simp]
theorem TopologicalSpace.Compacts.coe_finset_sup {α : Type u_1} [TopologicalSpace α] {ι : Type u_4} {s : Finset ι} {f : ιTopologicalSpace.Compacts α} :
(s.sup f) = s.sup fun (i : ι) => (f i)

The image of a compact set under a continuous function.

Equations
@[simp]
theorem TopologicalSpace.Compacts.coe_map {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : αβ} (hf : Continuous f) (s : TopologicalSpace.Compacts α) :

A homeomorphism induces an equivalence on compact sets, by taking the image.

Equations

The image of a compact set under a homeomorphism can also be expressed as a preimage.

The product of two TopologicalSpace.Compacts, as a TopologicalSpace.Compacts in the product space.

Equations
  • K.prod L = { carrier := K ×ˢ L, isCompact' := }
@[simp]
theorem TopologicalSpace.Compacts.coe_prod {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (K : TopologicalSpace.Compacts α) (L : TopologicalSpace.Compacts β) :
(K.prod L) = K ×ˢ L

Nonempty compact sets #

Equations

Reinterpret a nonempty compact as a closed set.

Equations
  • s.toCloseds = { carrier := s, closed' := }
@[simp]
theorem TopologicalSpace.NonemptyCompacts.coe_mk {α : Type u_1} [TopologicalSpace α] (s : TopologicalSpace.Compacts α) (h : s.carrier.Nonempty) :
{ toCompacts := s, nonempty' := h } = s
Equations
Equations
  • TopologicalSpace.NonemptyCompacts.instTopOfCompactSpaceOfNonempty = { top := { toCompacts := , nonempty' := } }
Equations
Equations
  • TopologicalSpace.NonemptyCompacts.instOrderTopOfCompactSpaceOfNonempty = OrderTop.lift SetLike.coe
@[simp]

In an inhabited space, the type of nonempty compact subsets is also inhabited, with default element the singleton set containing the default element.

Equations
  • TopologicalSpace.NonemptyCompacts.instInhabited = { default := { carrier := {default}, isCompact' := , nonempty' := } }

The product of two TopologicalSpace.NonemptyCompacts, as a TopologicalSpace.NonemptyCompacts in the product space.

Equations
  • K.prod L = { toCompacts := K.prod L.toCompacts, nonempty' := }

Positive compact sets #

Equations

Reinterpret a positive compact as a nonempty compact.

Equations
  • s.toNonemptyCompacts = { toCompacts := s.toCompacts, nonempty' := }
@[simp]
theorem TopologicalSpace.PositiveCompacts.coe_mk {α : Type u_1} [TopologicalSpace α] (s : TopologicalSpace.Compacts α) (h : (interior s.carrier).Nonempty) :
{ toCompacts := s, interior_nonempty' := h } = s
Equations
Equations
  • TopologicalSpace.PositiveCompacts.instTopOfCompactSpaceOfNonempty = { top := { toCompacts := , interior_nonempty' := } }
Equations
Equations
  • TopologicalSpace.PositiveCompacts.instOrderTopOfCompactSpaceOfNonempty = OrderTop.lift SetLike.coe
@[simp]

The image of a positive compact set under a continuous open map.

Equations
@[simp]
theorem TopologicalSpace.PositiveCompacts.coe_map {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : αβ} (hf : Continuous f) (hf' : IsOpenMap f) (s : TopologicalSpace.PositiveCompacts α) :
theorem exists_positiveCompacts_subset {α : Type u_1} [TopologicalSpace α] [LocallyCompactSpace α] {U : Set α} (ho : IsOpen U) (hn : U.Nonempty) :
theorem IsOpen.exists_positiveCompacts_closure_subset {α : Type u_1} [TopologicalSpace α] [R1Space α] [LocallyCompactSpace α] {U : Set α} (ho : IsOpen U) (hn : U.Nonempty) :
Equations
  • TopologicalSpace.PositiveCompacts.instInhabitedOfCompactSpaceOfNonempty = { default := }

In a nonempty locally compact space, there exists a compact set with nonempty interior.

Equations
  • =

The product of two TopologicalSpace.PositiveCompacts, as a TopologicalSpace.PositiveCompacts in the product space.

Equations
  • K.prod L = { toCompacts := K.prod L.toCompacts, interior_nonempty' := }

Compact open sets #

Equations

Reinterpret a compact open as an open.

Equations
  • s.toOpens = { carrier := s, is_open' := }
@[simp]

Reinterpret a compact open as a clopen.

Equations
  • s.toClopens = { carrier := s, isClopen' := }
@[simp]
theorem TopologicalSpace.CompactOpens.coe_mk {α : Type u_1} [TopologicalSpace α] (s : TopologicalSpace.Compacts α) (h : IsOpen s.carrier) :
{ toCompacts := s, isOpen' := h } = s
Equations
Equations
  • TopologicalSpace.CompactOpens.instBot = { bot := { toCompacts := , isOpen' := } }
Equations
Equations
  • TopologicalSpace.CompactOpens.instOrderBot = OrderBot.lift SetLike.coe
Equations
  • TopologicalSpace.CompactOpens.instInhabited = { default := }
Equations
Equations
Equations
  • TopologicalSpace.CompactOpens.instSDiff = { sdiff := fun (s t : TopologicalSpace.CompactOpens α) => { carrier := s \ t, isCompact' := , isOpen' := } }
Equations
Equations
  • TopologicalSpace.CompactOpens.instTop = { top := { toCompacts := , isOpen' := } }
@[simp]
Equations
  • TopologicalSpace.CompactOpens.instBoundedOrder = BoundedOrder.lift SetLike.coe
Equations
Equations
Equations
@[simp]
theorem TopologicalSpace.CompactOpens.map_toCompacts {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : αβ) (hf : Continuous f) (hf' : IsOpenMap f) (s : TopologicalSpace.CompactOpens α) :
(TopologicalSpace.CompactOpens.map f hf hf' s).toCompacts = TopologicalSpace.Compacts.map f hf s.toCompacts

The image of a compact open under a continuous open map.

Equations
@[simp]
theorem TopologicalSpace.CompactOpens.coe_map {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : αβ} (hf : Continuous f) (hf' : IsOpenMap f) (s : TopologicalSpace.CompactOpens α) :
(TopologicalSpace.CompactOpens.map f hf hf' s) = f '' s
theorem TopologicalSpace.CompactOpens.map_comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (f : βγ) (g : αβ) (hf : Continuous f) (hg : Continuous g) (hf' : IsOpenMap f) (hg' : IsOpenMap g) (K : TopologicalSpace.CompactOpens α) :

The product of two TopologicalSpace.CompactOpens, as a TopologicalSpace.CompactOpens in the product space.

Equations
  • K.prod L = { toCompacts := K.prod L.toCompacts, isOpen' := }
@[simp]