# 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,

• TopologicalSpace.Compacts α: The type of compact sets.
• TopologicalSpace.NonemptyCompacts α: The type of non-empty compact sets.
• TopologicalSpace.PositiveCompacts α: The type of compact sets with non-empty interior.
• TopologicalSpace.CompactOpens α: The type of compact open sets. This is a central object in the study of spectral spaces.

### Compact sets #

structure TopologicalSpace.Compacts (α : Type u_4) [] :
Type u_4

The type of compact sets of a topological space.

Instances For
def TopologicalSpace.Compacts.Simps.coe {α : Type u_1} [] (s : ) :
Set α

See Note [custom simps projection].

Instances For
theorem TopologicalSpace.Compacts.isCompact {α : Type u_1} [] (s : ) :
theorem TopologicalSpace.Compacts.ext {α : Type u_1} [] {s : } {t : } (h : s = t) :
s = t
@[simp]
theorem TopologicalSpace.Compacts.coe_mk {α : Type u_1} [] (s : Set α) (h : ) :
{ carrier := s, isCompact' := h } = s
@[simp]
theorem TopologicalSpace.Compacts.carrier_eq_coe {α : Type u_1} [] (s : ) :
s.carrier = s

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

@[simp]
theorem TopologicalSpace.Compacts.coe_sup {α : Type u_1} [] (s : ) (t : ) :
↑(s t) = s t
@[simp]
theorem TopologicalSpace.Compacts.coe_inf {α : Type u_1} [] [] (s : ) (t : ) :
↑(s t) = s t
@[simp]
theorem TopologicalSpace.Compacts.coe_top {α : Type u_1} [] [] :
= Set.univ
@[simp]
theorem TopologicalSpace.Compacts.coe_bot {α : Type u_1} [] :
=
@[simp]
theorem TopologicalSpace.Compacts.coe_finset_sup {α : Type u_1} [] {ι : Type u_4} {s : } {f : } :
↑() = Finset.sup s fun i => ↑(f i)
def TopologicalSpace.Compacts.map {α : Type u_1} {β : Type u_2} [] [] (f : αβ) (hf : ) (K : ) :

The image of a compact set under a continuous function.

Instances For
@[simp]
theorem TopologicalSpace.Compacts.coe_map {α : Type u_1} {β : Type u_2} [] [] {f : αβ} (hf : ) (s : ) :
↑() = f '' s
@[simp]
theorem TopologicalSpace.Compacts.map_id {α : Type u_1} [] (K : ) :
theorem TopologicalSpace.Compacts.map_comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} [] [] [] (f : βγ) (g : αβ) (hf : ) (hg : ) (K : ) :
@[simp]
theorem TopologicalSpace.Compacts.equiv_apply {α : Type u_1} {β : Type u_2} [] [] (f : α ≃ₜ β) (K : ) :
@[simp]
theorem TopologicalSpace.Compacts.equiv_symm_apply {α : Type u_1} {β : Type u_2} [] [] (f : α ≃ₜ β) (K : ) :
().symm K = TopologicalSpace.Compacts.map ↑() (_ : ) K
def TopologicalSpace.Compacts.equiv {α : Type u_1} {β : Type u_2} [] [] (f : α ≃ₜ β) :

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

Instances For
@[simp]
@[simp]
theorem TopologicalSpace.Compacts.equiv_trans {α : Type u_1} {β : Type u_2} {γ : Type u_3} [] [] [] (f : α ≃ₜ β) (g : β ≃ₜ γ) :
@[simp]
theorem TopologicalSpace.Compacts.equiv_symm {α : Type u_1} {β : Type u_2} [] [] (f : α ≃ₜ β) :
theorem TopologicalSpace.Compacts.coe_equiv_apply_eq_preimage {α : Type u_1} {β : Type u_2} [] [] (f : α ≃ₜ β) (K : ) :
↑() = ↑() ⁻¹' K

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

def TopologicalSpace.Compacts.prod {α : Type u_1} {β : Type u_2} [] [] (K : ) (L : ) :

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

Instances For
@[simp]
theorem TopologicalSpace.Compacts.coe_prod {α : Type u_1} {β : Type u_2} [] [] (K : ) (L : ) :
= K ×ˢ L

### Nonempty compact sets #

structure TopologicalSpace.NonemptyCompacts (α : Type u_4) [] extends :
Type u_4

The type of nonempty compact sets of a topological space.

Instances For

See Note [custom simps projection].

Instances For

Reinterpret a nonempty compact as a closed set.

Instances For
theorem TopologicalSpace.NonemptyCompacts.ext {α : Type u_1} [] (h : s = t) :
s = t
@[simp]
theorem TopologicalSpace.NonemptyCompacts.coe_mk {α : Type u_1} [] (s : ) (h : Set.Nonempty s.carrier) :
{ toCompacts := s, nonempty' := h } = s
theorem TopologicalSpace.NonemptyCompacts.carrier_eq_coe {α : Type u_1} [] :
s.carrier = s
@[simp]
theorem TopologicalSpace.NonemptyCompacts.coe_toCompacts {α : Type u_1} [] :
s.toCompacts = s
@[simp]
theorem TopologicalSpace.NonemptyCompacts.coe_sup {α : Type u_1} [] :
↑(s t) = s t
@[simp]
theorem TopologicalSpace.NonemptyCompacts.coe_top {α : Type u_1} [] [] [] :
= Set.univ

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

def TopologicalSpace.NonemptyCompacts.prod {α : Type u_1} {β : Type u_2} [] [] :

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

Instances For
@[simp]
theorem TopologicalSpace.NonemptyCompacts.coe_prod {α : Type u_1} {β : Type u_2} [] [] :
= K ×ˢ L

### Positive compact sets #

structure TopologicalSpace.PositiveCompacts (α : Type u_4) [] extends :
Type u_4

The type of compact sets with nonempty interior of a topological space. See also TopologicalSpace.Compacts and TopologicalSpace.NonemptyCompacts.

Instances For

See Note [custom simps projection].

Instances For

Reinterpret a positive compact as a nonempty compact.

Instances For
theorem TopologicalSpace.PositiveCompacts.ext {α : Type u_1} [] (h : s = t) :
s = t
@[simp]
theorem TopologicalSpace.PositiveCompacts.coe_mk {α : Type u_1} [] (s : ) (h : Set.Nonempty (interior s.carrier)) :
{ toCompacts := s, interior_nonempty' := h } = s
theorem TopologicalSpace.PositiveCompacts.carrier_eq_coe {α : Type u_1} [] :
s.carrier = s
@[simp]
theorem TopologicalSpace.PositiveCompacts.coe_toCompacts {α : Type u_1} [] :
s.toCompacts = s
@[simp]
theorem TopologicalSpace.PositiveCompacts.coe_sup {α : Type u_1} [] :
↑(s t) = s t
@[simp]
theorem TopologicalSpace.PositiveCompacts.coe_top {α : Type u_1} [] [] [] :
= Set.univ
def TopologicalSpace.PositiveCompacts.map {α : Type u_1} {β : Type u_2} [] [] (f : αβ) (hf : ) (hf' : ) :

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

Instances For
@[simp]
theorem TopologicalSpace.PositiveCompacts.coe_map {α : Type u_1} {β : Type u_2} [] [] {f : αβ} (hf : ) (hf' : ) :
↑() = f '' s
theorem TopologicalSpace.PositiveCompacts.map_comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} [] [] [] (f : βγ) (g : αβ) (hf : ) (hg : ) (hf' : ) (hg' : ) :
theorem exists_positiveCompacts_subset {α : Type u_1} [] {U : Set α} (ho : ) (hn : ) :
K, K U

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

def TopologicalSpace.PositiveCompacts.prod {α : Type u_1} {β : Type u_2} [] [] :

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

Instances For
@[simp]
theorem TopologicalSpace.PositiveCompacts.coe_prod {α : Type u_1} {β : Type u_2} [] [] :
= K ×ˢ L

### Compact open sets #

structure TopologicalSpace.CompactOpens (α : Type u_4) [] extends :
Type u_4

The type of compact open sets of a topological space. This is useful in non Hausdorff contexts, in particular spectral spaces.

Instances For

See Note [custom simps projection].

Instances For

Reinterpret a compact open as an open.

Instances For
@[simp]

Reinterpret a compact open as a clopen.

Instances For
theorem TopologicalSpace.CompactOpens.ext {α : Type u_1} [] (h : s = t) :
s = t
@[simp]
theorem TopologicalSpace.CompactOpens.coe_mk {α : Type u_1} [] (s : ) (h : IsOpen s.carrier) :
{ toCompacts := s, isOpen' := h } = s
@[simp]
theorem TopologicalSpace.CompactOpens.coe_sup {α : Type u_1} [] :
↑(s t) = s t
@[simp]
theorem TopologicalSpace.CompactOpens.coe_inf {α : Type u_1} [] [] :
↑(s t) = s t
@[simp]
theorem TopologicalSpace.CompactOpens.coe_top {α : Type u_1} [] [] :
= Set.univ
@[simp]
@[simp]
theorem TopologicalSpace.CompactOpens.coe_sdiff {α : Type u_1} [] [] :
↑(s \ t) = s \ t
@[simp]
theorem TopologicalSpace.CompactOpens.coe_compl {α : Type u_1} [] [] [] :
s = (s)
@[simp]
theorem TopologicalSpace.CompactOpens.map_toCompacts {α : Type u_1} {β : Type u_2} [] [] (f : αβ) (hf : ) (hf' : ) :
().toCompacts = TopologicalSpace.Compacts.map f hf s.toCompacts
def TopologicalSpace.CompactOpens.map {α : Type u_1} {β : Type u_2} [] [] (f : αβ) (hf : ) (hf' : ) :

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

Instances For
@[simp]
theorem TopologicalSpace.CompactOpens.coe_map {α : Type u_1} {β : Type u_2} [] [] {f : αβ} (hf : ) (hf' : ) :
↑() = f '' s
@[simp]
theorem TopologicalSpace.CompactOpens.map_comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} [] [] [] (f : βγ) (g : αβ) (hf : ) (hg : ) (hf' : ) (hg' : ) :
def TopologicalSpace.CompactOpens.prod {α : Type u_1} {β : Type u_2} [] [] :

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

Instances For
@[simp]
theorem TopologicalSpace.CompactOpens.coe_prod {α : Type u_1} {β : Type u_2} [] [] :
= K ×ˢ L