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

  the carrier set, i.e. the points in this set

• closed' : IsClosed self.carrier

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

Equations

• TopologicalSpace.Closeds.instSetLike = { coe := TopologicalSpace.Closeds.carrier, coe_injective' := ⋯ }

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

Equations

• ⋯ = ⋯

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 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

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' := ⋯ }

@[simp]

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

@[simp]

theorem TopologicalSpace.Closeds.mem_closure {α : Type u_2} [TopologicalSpace α] {s : Set α} {x : α} : x ∈ TopologicalSpace.Closeds.closure s ↔ x ∈ 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

• TopologicalSpace.Closeds.gi = { choice := fun (s : Set α) (hs : ↑(TopologicalSpace.Closeds.closure s) ≤ s) => { carrier := s, closed' := ⋯ }, gc := ⋯, le_l_u := ⋯, choice_eq := ⋯ }

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.instInhabited {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.instInhabited = { default := ⊥ }

@[simp]

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

@[simp]

theorem TopologicalSpace.Closeds.coe_inf {α : Type u_2} [TopologicalSpace α] (s 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} : (↑s).Nonempty ↔ 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) : ↑(s.sup f) = s.sup (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) : ↑(s.inf f) = s.inf (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' := ⋯ }

@[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' := ⋯ } = { carrier := ⋂ (i : ι), s i, closed' := ⋯ }

def TopologicalSpace.Closeds.coframeMinimalAxioms {α : Type u_2} [TopologicalSpace α] : Order.Coframe.MinimalAxioms (TopologicalSpace.Closeds α)

Closed sets in a topological space form a coframe.

Equations

• TopologicalSpace.Closeds.coframeMinimalAxioms = Order.Coframe.MinimalAxioms.mk ⋯

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

Equations

• TopologicalSpace.Closeds.instCoframe = Order.Coframe.ofMinimalAxioms TopologicalSpace.Closeds.coframeMinimalAxioms

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' := ⋯ }

@[simp]

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

@[simp]

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

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

• s.compl = { carrier := (↑s)ᶜ, is_open' := ⋯ }

@[simp]

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

• s.compl = { carrier := (↑s)ᶜ, closed' := ⋯ }

@[simp]

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

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

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

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.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.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.

@[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✝

@[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✝

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.singleton x).compl

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

  the carrier set, i.e. the points in this set

• isClopen' : IsClopen self.carrier

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

Equations

• TopologicalSpace.Clopens.instSetLike = { coe := fun (s : TopologicalSpace.Clopens α) => s.carrier, coe_injective' := ⋯ }

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

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

Reinterpret a clopen as an open.

Equations

• s.toOpens = { carrier := ↑s, is_open' := ⋯ }

@[simp]

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

theorem TopologicalSpace.Clopens.ext {α : Type u_2} [TopologicalSpace α] {s 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.instMax {α : Type u_2} [TopologicalSpace α] : Max (TopologicalSpace.Clopens α)

Equations

• TopologicalSpace.Clopens.instMax = { max := fun (s t : TopologicalSpace.Clopens α) => { carrier := ↑s ∪ ↑t, isClopen' := ⋯ } }

instance TopologicalSpace.Clopens.instMin {α : Type u_2} [TopologicalSpace α] : Min (TopologicalSpace.Clopens α)

Equations

• TopologicalSpace.Clopens.instMin = { min := fun (s t : TopologicalSpace.Clopens α) => { carrier := ↑s ∩ ↑t, isClopen' := ⋯ } }

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

Equations

• TopologicalSpace.Clopens.instTop = { top := { carrier := ⊤, isClopen' := ⋯ } }

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

Equations

• TopologicalSpace.Clopens.instBot = { bot := { carrier := ⊥, isClopen' := ⋯ } }

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

Equations

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

instance TopologicalSpace.Clopens.instHImp {α : Type u_2} [TopologicalSpace α] : HImp (TopologicalSpace.Clopens α)

Equations

• TopologicalSpace.Clopens.instHImp = { himp := fun (s t : TopologicalSpace.Clopens α) => { carrier := ↑s ⇨ ↑t, isClopen' := ⋯ } }

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

Equations

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

@[simp]

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

@[simp]

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

@[simp]

theorem TopologicalSpace.Clopens.coe_himp {α : Type u_2} [TopologicalSpace α] (s 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.instBooleanAlgebra {α : Type u_2} [TopologicalSpace α] : BooleanAlgebra (TopologicalSpace.Clopens α)

Equations

• TopologicalSpace.Clopens.instBooleanAlgebra = Function.Injective.booleanAlgebra SetLike.coe ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

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

Equations

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

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

Equations

• TopologicalSpace.Clopens.instSProdProd = { sprod := fun (s : TopologicalSpace.Clopens α) (t : TopologicalSpace.Clopens β) => { carrier := ↑s ×ˢ ↑t, isClopen' := ⋯ } }

@[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

Irreducible closed sets

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

The type of irreducible closed subsets of a topological space.

• carrier : Set α

  the carrier set, i.e. the points in this set

• is_irreducible' : IsIrreducible self.carrier
• is_closed' : IsClosed self.carrier

instance TopologicalSpace.IrreducibleCloseds.instSetLike {α : Type u_2} [TopologicalSpace α] : SetLike (TopologicalSpace.IrreducibleCloseds α) α

Equations

• TopologicalSpace.IrreducibleCloseds.instSetLike = { coe := TopologicalSpace.IrreducibleCloseds.carrier, coe_injective' := ⋯ }

instance TopologicalSpace.IrreducibleCloseds.instCanLiftSetCoeAndIsIrreducibleIsClosed {α : Type u_2} [TopologicalSpace α] : CanLift (Set α) (TopologicalSpace.IrreducibleCloseds α) SetLike.coe fun (s : Set α) => IsIrreducible s ∧ IsClosed s

Equations

• ⋯ = ⋯

theorem TopologicalSpace.IrreducibleCloseds.isIrreducible {α : Type u_2} [TopologicalSpace α] (s : TopologicalSpace.IrreducibleCloseds α) : IsIrreducible ↑s

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

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

See Note [custom simps projection].

Equations

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

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

@[simp]

theorem TopologicalSpace.IrreducibleCloseds.coe_mk {α : Type u_2} [TopologicalSpace α] (s : Set α) (h : IsIrreducible s) (h' : IsClosed s) : ↑{ carrier := s, is_irreducible' := h, is_closed' := h' } = s

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

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

Equations

• TopologicalSpace.IrreducibleCloseds.singleton x = { carrier := {x}, is_irreducible' := ⋯, is_closed' := ⋯ }

@[simp]

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

@[simp]

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

def TopologicalSpace.IrreducibleCloseds.equivSubtype {α : Type u_2} [TopologicalSpace α] : TopologicalSpace.IrreducibleCloseds α ≃ { x : Set α // IsIrreducible x ∧ IsClosed x }

The equivalence between IrreducibleCloseds α and {x : Set α // IsIrreducible x ∧ IsClosed x }.

Equations

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

@[simp]

theorem TopologicalSpace.IrreducibleCloseds.equivSubtype_symm_apply {α : Type u_2} [TopologicalSpace α] (a : { x : Set α // IsIrreducible x ∧ IsClosed x }) : TopologicalSpace.IrreducibleCloseds.equivSubtype.symm a = { carrier := ↑a, is_irreducible' := ⋯, is_closed' := ⋯ }

@[simp]

theorem TopologicalSpace.IrreducibleCloseds.equivSubtype_apply {α : Type u_2} [TopologicalSpace α] (a : TopologicalSpace.IrreducibleCloseds α) : TopologicalSpace.IrreducibleCloseds.equivSubtype a = ⟨a.carrier, ⋯⟩

def TopologicalSpace.IrreducibleCloseds.equivSubtype' {α : Type u_2} [TopologicalSpace α] : TopologicalSpace.IrreducibleCloseds α ≃ { x : Set α // IsClosed x ∧ IsIrreducible x }

The equivalence between IrreducibleCloseds α and {x : Set α // IsClosed x ∧ IsIrreducible x }.

Equations

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

@[simp]

theorem TopologicalSpace.IrreducibleCloseds.equivSubtype'_apply {α : Type u_2} [TopologicalSpace α] (a : TopologicalSpace.IrreducibleCloseds α) : TopologicalSpace.IrreducibleCloseds.equivSubtype' a = ⟨a.carrier, ⋯⟩

@[simp]

theorem TopologicalSpace.IrreducibleCloseds.equivSubtype'_symm_apply {α : Type u_2} [TopologicalSpace α] (a : { x : Set α // IsClosed x ∧ IsIrreducible x }) : TopologicalSpace.IrreducibleCloseds.equivSubtype'.symm a = { carrier := ↑a, is_irreducible' := ⋯, is_closed' := ⋯ }

def TopologicalSpace.IrreducibleCloseds.orderIsoSubtype (α : Type u_2) [TopologicalSpace α] : TopologicalSpace.IrreducibleCloseds α ≃o { x : Set α // IsIrreducible x ∧ IsClosed x }

The equivalence IrreducibleCloseds α ≃ { x : Set α // IsIrreducible x ∧ IsClosed x } is an order isomorphism.

Equations

• TopologicalSpace.IrreducibleCloseds.orderIsoSubtype α = TopologicalSpace.IrreducibleCloseds.equivSubtype.toOrderIso ⋯ ⋯

def TopologicalSpace.IrreducibleCloseds.orderIsoSubtype' (α : Type u_2) [TopologicalSpace α] : TopologicalSpace.IrreducibleCloseds α ≃o { x : Set α // IsClosed x ∧ IsIrreducible x }

The equivalence IrreducibleCloseds α ≃ { x : Set α // IsClosed x ∧ IsIrreducible x } is an order isomorphism.

Equations

• TopologicalSpace.IrreducibleCloseds.orderIsoSubtype' α = TopologicalSpace.IrreducibleCloseds.equivSubtype'.toOrderIso ⋯ ⋯