Closed sets

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

Main Definitions

For a topological space α,

• TopologicalSpace.Closeds α: The type of closed sets.
• TopologicalSpace.Clopens α: The type of clopen sets.

Closed sets

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

The type of closed subsets of a topological space.

• carrier : Set α
• closed' : IsClosed self.carrier

instance TopologicalSpace.Closeds.instSetLikeCloseds {α : Type u_2} [TopologicalSpace α] : SetLike (TopologicalSpace.Closeds α) α

Equations

• TopologicalSpace.Closeds.instSetLikeCloseds = { coe := TopologicalSpace.Closeds.carrier, coe_injective' := (_ : ∀ (s t : TopologicalSpace.Closeds α), s.carrier = t.carrier → s = t) }

instance TopologicalSpace.Closeds.instCanLiftSetClosedsCoeInstSetLikeClosedsIsClosed {α : Type u_2} [TopologicalSpace α] : CanLift (Set α) (TopologicalSpace.Closeds α) SetLike.coe IsClosed

Equations

• (_ : CanLift (Set α) (TopologicalSpace.Closeds α) SetLike.coe IsClosed) = (_ : CanLift (Set α) (TopologicalSpace.Closeds α) SetLike.coe IsClosed)

theorem TopologicalSpace.Closeds.closed {α : Type u_2} [TopologicalSpace α] (s : TopologicalSpace.Closeds α) : IsClosed ↑s

def TopologicalSpace.Closeds.Simps.coe {α : Type u_2} [TopologicalSpace α] (s : TopologicalSpace.Closeds α) : Set α

See Note [custom simps projection].

Equations

• TopologicalSpace.Closeds.Simps.coe s = ↑s

theorem TopologicalSpace.Closeds.ext {α : Type u_2} [TopologicalSpace α] {s : TopologicalSpace.Closeds α} {t : TopologicalSpace.Closeds α} (h : ↑s = ↑t) : s = t

@[simp]

theorem TopologicalSpace.Closeds.coe_mk {α : Type u_2} [TopologicalSpace α] (s : Set α) (h : IsClosed s) : ↑{ carrier := s, closed' := h } = s

@[simp]

theorem TopologicalSpace.Closeds.coe_closure {α : Type u_2} [TopologicalSpace α] (s : Set α) : ↑(TopologicalSpace.Closeds.closure s) = closure s

def TopologicalSpace.Closeds.closure {α : Type u_2} [TopologicalSpace α] (s : Set α) : TopologicalSpace.Closeds α

The closure of a set, as an element of TopologicalSpace.Closeds.

Equations

• TopologicalSpace.Closeds.closure s = { carrier := closure s, closed' := (_ : IsClosed (closure s)) }

theorem TopologicalSpace.Closeds.gc {α : Type u_2} [TopologicalSpace α] : GaloisConnection TopologicalSpace.Closeds.closure SetLike.coe

def TopologicalSpace.Closeds.gi {α : Type u_2} [TopologicalSpace α] : GaloisInsertion TopologicalSpace.Closeds.closure SetLike.coe

The galois coinsertion between sets and opens.

Equations

• One or more equations did not get rendered due to their size.

instance TopologicalSpace.Closeds.completeLattice {α : Type u_2} [TopologicalSpace α] : CompleteLattice (TopologicalSpace.Closeds α)

Equations

• One or more equations did not get rendered due to their size.

instance TopologicalSpace.Closeds.instInhabitedCloseds {α : Type u_2} [TopologicalSpace α] : Inhabited (TopologicalSpace.Closeds α)

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

Equations

• TopologicalSpace.Closeds.instInhabitedCloseds = { default := ⊥ }

@[simp]

theorem TopologicalSpace.Closeds.coe_sup {α : Type u_2} [TopologicalSpace α] (s : TopologicalSpace.Closeds α) (t : TopologicalSpace.Closeds α) : ↑(s ⊔ t) = ↑s ∪ ↑t

@[simp]

theorem TopologicalSpace.Closeds.coe_inf {α : Type u_2} [TopologicalSpace α] (s : TopologicalSpace.Closeds α) (t : TopologicalSpace.Closeds α) : ↑(s ⊓ t) = ↑s ∩ ↑t

@[simp]

theorem TopologicalSpace.Closeds.coe_top {α : Type u_2} [TopologicalSpace α] : ↑⊤ = Set.univ

@[simp]

theorem TopologicalSpace.Closeds.coe_eq_univ {α : Type u_2} [TopologicalSpace α] {s : TopologicalSpace.Closeds α} : ↑s = Set.univ ↔ s = ⊤

@[simp]

theorem TopologicalSpace.Closeds.coe_bot {α : Type u_2} [TopologicalSpace α] : ↑⊥ = ∅

@[simp]

theorem TopologicalSpace.Closeds.coe_eq_empty {α : Type u_2} [TopologicalSpace α] {s : TopologicalSpace.Closeds α} : ↑s = ∅ ↔ s = ⊥

theorem TopologicalSpace.Closeds.coe_nonempty {α : Type u_2} [TopologicalSpace α] {s : TopologicalSpace.Closeds α} : Set.Nonempty ↑s ↔ s ≠ ⊥

@[simp]

theorem TopologicalSpace.Closeds.coe_sInf {α : Type u_2} [TopologicalSpace α] {S : Set (TopologicalSpace.Closeds α)} : ↑(sInf S) = ⋂ i ∈ S, ↑i

@[simp]

theorem TopologicalSpace.Closeds.coe_sSup {α : Type u_2} [TopologicalSpace α] {S : Set (TopologicalSpace.Closeds α)} : ↑(sSup S) = closure (⋃₀ (SetLike.coe '' S))

@[simp]

theorem TopologicalSpace.Closeds.coe_finset_sup {ι : Type u_1} {α : Type u_2} [TopologicalSpace α] (f : ι → TopologicalSpace.Closeds α) (s : Finset ι) : ↑(Finset.sup s f) = Finset.sup s (SetLike.coe ∘ f)

@[simp]

theorem TopologicalSpace.Closeds.coe_finset_inf {ι : Type u_1} {α : Type u_2} [TopologicalSpace α] (f : ι → TopologicalSpace.Closeds α) (s : Finset ι) : ↑(Finset.inf s f) = Finset.inf s (SetLike.coe ∘ f)

@[simp]

theorem TopologicalSpace.Closeds.mem_sInf {α : Type u_2} [TopologicalSpace α] {S : Set (TopologicalSpace.Closeds α)} {x : α} : x ∈ sInf S ↔ ∀ s ∈ S, x ∈ s

@[simp]

theorem TopologicalSpace.Closeds.mem_iInf {α : Type u_2} [TopologicalSpace α] {ι : Sort u_4} {x : α} {s : ι → TopologicalSpace.Closeds α} : x ∈ iInf s ↔ ∀ (i : ι), x ∈ s i

@[simp]

theorem TopologicalSpace.Closeds.coe_iInf {α : Type u_2} [TopologicalSpace α] {ι : Sort u_4} (s : ι → TopologicalSpace.Closeds α) : ↑(⨅ (i : ι), s i) = ⋂ (i : ι), ↑(s i)

theorem TopologicalSpace.Closeds.iInf_def {α : Type u_2} [TopologicalSpace α] {ι : Sort u_4} (s : ι → TopologicalSpace.Closeds α) : ⨅ (i : ι), s i = { carrier := ⋂ (i : ι), ↑(s i), closed' := (_ : IsClosed (⋂ (i : ι), ↑(s i))) }

@[simp]

theorem TopologicalSpace.Closeds.iInf_mk {α : Type u_2} [TopologicalSpace α] {ι : Sort u_4} (s : ι → Set α) (h : ∀ (i : ι), IsClosed (s i)) : ⨅ (i : ι), { carrier := s i, closed' := (_ : IsClosed (s i)) } = { carrier := ⋂ (i : ι), s i, closed' := (_ : IsClosed (⋂ (i : ι), s i)) }

instance TopologicalSpace.Closeds.instCoframeCloseds {α : Type u_2} [TopologicalSpace α] : Order.Coframe (TopologicalSpace.Closeds α)

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem TopologicalSpace.Closeds.coe_singleton {α : Type u_2} [TopologicalSpace α] [T1Space α] (x : α) : ↑(TopologicalSpace.Closeds.singleton x) = {x}

def TopologicalSpace.Closeds.singleton {α : Type u_2} [TopologicalSpace α] [T1Space α] (x : α) : TopologicalSpace.Closeds α

The term of TopologicalSpace.Closeds α corresponding to a singleton.

Equations

• TopologicalSpace.Closeds.singleton x = { carrier := {x}, closed' := (_ : IsClosed {x}) }

@[simp]

theorem TopologicalSpace.Closeds.mem_singleton {α : Type u_2} [TopologicalSpace α] [T1Space α] {a : α} {b : α} : a ∈ TopologicalSpace.Closeds.singleton b ↔ a = b

@[simp]

theorem TopologicalSpace.Closeds.compl_coe {α : Type u_2} [TopologicalSpace α] (s : TopologicalSpace.Closeds α) : ↑(TopologicalSpace.Closeds.compl s) = (↑s)ᶜ

def TopologicalSpace.Closeds.compl {α : Type u_2} [TopologicalSpace α] (s : TopologicalSpace.Closeds α) : TopologicalSpace.Opens α

The complement of a closed set as an open set.

Equations

• TopologicalSpace.Closeds.compl s = { carrier := (↑s)ᶜ, is_open' := (_ : IsOpen s.carrierᶜ) }

@[simp]

theorem TopologicalSpace.Opens.coe_compl {α : Type u_2} [TopologicalSpace α] (s : TopologicalSpace.Opens α) : ↑(TopologicalSpace.Opens.compl s) = (↑s)ᶜ

def TopologicalSpace.Opens.compl {α : Type u_2} [TopologicalSpace α] (s : TopologicalSpace.Opens α) : TopologicalSpace.Closeds α

The complement of an open set as a closed set.

Equations

• TopologicalSpace.Opens.compl s = { carrier := (↑s)ᶜ, closed' := (_ : IsClosed s.carrierᶜ) }

theorem TopologicalSpace.Closeds.compl_compl {α : Type u_2} [TopologicalSpace α] (s : TopologicalSpace.Closeds α) : TopologicalSpace.Opens.compl (TopologicalSpace.Closeds.compl s) = s

theorem TopologicalSpace.Opens.compl_compl {α : Type u_2} [TopologicalSpace α] (s : TopologicalSpace.Opens α) : TopologicalSpace.Closeds.compl (TopologicalSpace.Opens.compl s) = s

theorem TopologicalSpace.Closeds.compl_bijective {α : Type u_2} [TopologicalSpace α] : Function.Bijective TopologicalSpace.Closeds.compl

theorem TopologicalSpace.Opens.compl_bijective {α : Type u_2} [TopologicalSpace α] : Function.Bijective TopologicalSpace.Opens.compl

@[simp]

theorem TopologicalSpace.Closeds.complOrderIso_symm_apply (α : Type u_2) [TopologicalSpace α] : ∀ (a : (TopologicalSpace.Opens α)ᵒᵈ), (RelIso.symm (TopologicalSpace.Closeds.complOrderIso α)) a = (TopologicalSpace.Opens.compl ∘ ⇑OrderDual.ofDual) a

@[simp]

theorem TopologicalSpace.Closeds.complOrderIso_apply (α : Type u_2) [TopologicalSpace α] : ∀ (a : TopologicalSpace.Closeds α), (TopologicalSpace.Closeds.complOrderIso α) a = (⇑OrderDual.toDual ∘ TopologicalSpace.Closeds.compl) a

def TopologicalSpace.Closeds.complOrderIso (α : Type u_2) [TopologicalSpace α] : TopologicalSpace.Closeds α ≃o (TopologicalSpace.Opens α)ᵒᵈ

TopologicalSpace.Closeds.compl as an OrderIso to the order dual of TopologicalSpace.Opens α.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem TopologicalSpace.Opens.complOrderIso_symm_apply (α : Type u_2) [TopologicalSpace α] : ∀ (a : (TopologicalSpace.Closeds α)ᵒᵈ), (RelIso.symm (TopologicalSpace.Opens.complOrderIso α)) a = (TopologicalSpace.Closeds.compl ∘ ⇑OrderDual.ofDual) a

@[simp]

theorem TopologicalSpace.Opens.complOrderIso_apply (α : Type u_2) [TopologicalSpace α] : ∀ (a : TopologicalSpace.Opens α), (TopologicalSpace.Opens.complOrderIso α) a = (⇑OrderDual.toDual ∘ TopologicalSpace.Opens.compl) a

def TopologicalSpace.Opens.complOrderIso (α : Type u_2) [TopologicalSpace α] : TopologicalSpace.Opens α ≃o (TopologicalSpace.Closeds α)ᵒᵈ

TopologicalSpace.Opens.compl as an OrderIso to the order dual of TopologicalSpace.Closeds α.

Equations

• One or more equations did not get rendered due to their size.

theorem TopologicalSpace.Closeds.coe_eq_singleton_of_isAtom {α : Type u_2} [TopologicalSpace α] [T0Space α] {s : TopologicalSpace.Closeds α} (hs : IsAtom s) : ∃ (a : α), ↑s = {a}

@[simp]

theorem TopologicalSpace.Closeds.isAtom_coe {α : Type u_2} [TopologicalSpace α] [T1Space α] {s : TopologicalSpace.Closeds α} : IsAtom ↑s ↔ IsAtom s

theorem TopologicalSpace.Closeds.isAtom_iff {α : Type u_2} [TopologicalSpace α] [T1Space α] {s : TopologicalSpace.Closeds α} : IsAtom s ↔ ∃ (x : α), s = TopologicalSpace.Closeds.singleton x

in a T1Space, atoms of TopologicalSpace.Closeds α are precisely the TopologicalSpace.Closeds.singletons.

theorem TopologicalSpace.Opens.isCoatom_iff {α : Type u_2} [TopologicalSpace α] [T1Space α] {s : TopologicalSpace.Opens α} : IsCoatom s ↔ ∃ (x : α), s = TopologicalSpace.Closeds.compl (TopologicalSpace.Closeds.singleton x)

in a T1Space, coatoms of TopologicalSpace.Opens α are precisely complements of singletons: (TopologicalSpace.Closeds.singleton x).compl.

Clopen sets

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

The type of clopen sets of a topological space.

• carrier : Set α
• isClopen' : IsClopen self.carrier

instance TopologicalSpace.Clopens.instSetLikeClopens {α : Type u_2} [TopologicalSpace α] : SetLike (TopologicalSpace.Clopens α) α

Equations

• One or more equations did not get rendered due to their size.

theorem TopologicalSpace.Clopens.isClopen {α : Type u_2} [TopologicalSpace α] (s : TopologicalSpace.Clopens α) : IsClopen ↑s

def TopologicalSpace.Clopens.Simps.coe {α : Type u_2} [TopologicalSpace α] (s : TopologicalSpace.Clopens α) : Set α

See Note [custom simps projection].

Equations

• TopologicalSpace.Clopens.Simps.coe s = ↑s

@[simp]

theorem TopologicalSpace.Clopens.toOpens_coe {α : Type u_2} [TopologicalSpace α] (s : TopologicalSpace.Clopens α) : ↑(TopologicalSpace.Clopens.toOpens s) = ↑s

def TopologicalSpace.Clopens.toOpens {α : Type u_2} [TopologicalSpace α] (s : TopologicalSpace.Clopens α) : TopologicalSpace.Opens α

Reinterpret a clopen as an open.

Equations

• TopologicalSpace.Clopens.toOpens s = { carrier := ↑s, is_open' := (_ : IsOpen ↑s) }

theorem TopologicalSpace.Clopens.ext {α : Type u_2} [TopologicalSpace α] {s : TopologicalSpace.Clopens α} {t : TopologicalSpace.Clopens α} (h : ↑s = ↑t) : s = t

@[simp]

theorem TopologicalSpace.Clopens.coe_mk {α : Type u_2} [TopologicalSpace α] (s : Set α) (h : IsClopen s) : ↑{ carrier := s, isClopen' := h } = s

@[simp]

theorem TopologicalSpace.Clopens.mem_mk {α : Type u_2} [TopologicalSpace α] {s : Set α} {x : α} {h : IsClopen s} : x ∈ { carrier := s, isClopen' := h } ↔ x ∈ s

instance TopologicalSpace.Clopens.instSupClopens {α : Type u_2} [TopologicalSpace α] : Sup (TopologicalSpace.Clopens α)

Equations

• TopologicalSpace.Clopens.instSupClopens = { sup := fun (s t : TopologicalSpace.Clopens α) => { carrier := ↑s ∪ ↑t, isClopen' := (_ : IsClopen (↑s ∪ ↑t)) } }

instance TopologicalSpace.Clopens.instInfClopens {α : Type u_2} [TopologicalSpace α] : Inf (TopologicalSpace.Clopens α)

Equations

• TopologicalSpace.Clopens.instInfClopens = { inf := fun (s t : TopologicalSpace.Clopens α) => { carrier := ↑s ∩ ↑t, isClopen' := (_ : IsClopen (↑s ∩ ↑t)) } }

instance TopologicalSpace.Clopens.instTopClopens {α : Type u_2} [TopologicalSpace α] : Top (TopologicalSpace.Clopens α)

Equations

• TopologicalSpace.Clopens.instTopClopens = { top := { carrier := ⊤, isClopen' := (_ : IsClopen Set.univ) } }

instance TopologicalSpace.Clopens.instBotClopens {α : Type u_2} [TopologicalSpace α] : Bot (TopologicalSpace.Clopens α)

Equations

• TopologicalSpace.Clopens.instBotClopens = { bot := { carrier := ⊥, isClopen' := (_ : IsClopen ∅) } }

instance TopologicalSpace.Clopens.instSDiffClopens {α : Type u_2} [TopologicalSpace α] : SDiff (TopologicalSpace.Clopens α)

Equations

• TopologicalSpace.Clopens.instSDiffClopens = { sdiff := fun (s t : TopologicalSpace.Clopens α) => { carrier := ↑s \ ↑t, isClopen' := (_ : IsClopen (↑s \ ↑t)) } }

instance TopologicalSpace.Clopens.instHasComplClopens {α : Type u_2} [TopologicalSpace α] : HasCompl (TopologicalSpace.Clopens α)

Equations

• TopologicalSpace.Clopens.instHasComplClopens = { compl := fun (s : TopologicalSpace.Clopens α) => { carrier := (↑s)ᶜ, isClopen' := (_ : IsClopen (↑s)ᶜ) } }

instance TopologicalSpace.Clopens.instBooleanAlgebraClopens {α : Type u_2} [TopologicalSpace α] : BooleanAlgebra (TopologicalSpace.Clopens α)

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem TopologicalSpace.Clopens.coe_sup {α : Type u_2} [TopologicalSpace α] (s : TopologicalSpace.Clopens α) (t : TopologicalSpace.Clopens α) : ↑(s ⊔ t) = ↑s ∪ ↑t

@[simp]

theorem TopologicalSpace.Clopens.coe_inf {α : Type u_2} [TopologicalSpace α] (s : TopologicalSpace.Clopens α) (t : TopologicalSpace.Clopens α) : ↑(s ⊓ t) = ↑s ∩ ↑t

@[simp]

theorem TopologicalSpace.Clopens.coe_top {α : Type u_2} [TopologicalSpace α] : ↑⊤ = Set.univ

@[simp]

theorem TopologicalSpace.Clopens.coe_bot {α : Type u_2} [TopologicalSpace α] : ↑⊥ = ∅

@[simp]

theorem TopologicalSpace.Clopens.coe_sdiff {α : Type u_2} [TopologicalSpace α] (s : TopologicalSpace.Clopens α) (t : TopologicalSpace.Clopens α) : ↑(s \ t) = ↑s \ ↑t

@[simp]

theorem TopologicalSpace.Clopens.coe_compl {α : Type u_2} [TopologicalSpace α] (s : TopologicalSpace.Clopens α) : ↑sᶜ = (↑s)ᶜ

instance TopologicalSpace.Clopens.instInhabitedClopens {α : Type u_2} [TopologicalSpace α] : Inhabited (TopologicalSpace.Clopens α)

Equations

• TopologicalSpace.Clopens.instInhabitedClopens = { default := ⊥ }

instance TopologicalSpace.Clopens.instSProdClopensClopensClopensProdInstTopologicalSpaceProd {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] : SProd (TopologicalSpace.Clopens α) (TopologicalSpace.Clopens β) (TopologicalSpace.Clopens (α × β))

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem TopologicalSpace.Clopens.mem_prod {α : Type u_2} {β : Type u_3} [TopologicalSpace α] [TopologicalSpace β] {s : TopologicalSpace.Clopens α} {t : TopologicalSpace.Clopens β} {x : α × β} : x ∈ s ×ˢ t ↔ x.1 ∈ s ∧ x.2 ∈ t