File: Mathlib/Topology/ExtremallyDisconnected.lean

full_name: ExtremallyDisconnected.open_closure
code: /-- An extremally disconnected topological space is a space
in which the closure of every open set is open. -/
class ExtremallyDisconnected : Prop where
  /-- The closure of every open set is open. -/
  open_closure : ∀ U : Set X, IsOpen U → IsOpen (closure U)
goal: IsOpen U → IsOpen (closure U)

full_name: ExtremallyDisconnected
code: /-- An extremally disconnected topological space is a space
in which the closure of every open set is open. -/
class ExtremallyDisconnected : Prop where
  /-- The closure of every open set is open. -/
  open_closure : ∀ U : Set X, IsOpen U → IsOpen (closure U)
goal: Prop

full_name: extremallyDisconnected_of_homeo
code: theorem extremallyDisconnected_of_homeo {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
    [ExtremallyDisconnected X] (e : X ≃ₜ Y) : ExtremallyDisconnected Y 
goal: ExtremallyDisconnected Y

full_name: instTotallySeparatedSpaceOfExtremallyDisconnectedOfT2Space
code: /-- Extremally disconnected spaces are totally separated. -/
instance [ExtremallyDisconnected X] [T2Space X] : TotallySeparatedSpace X 
goal: TotallySeparatedSpace X

full_name: CompactT2.Projective
code: /-- The assertion `CompactT2.Projective` states that given continuous maps
`f : X → Z` and `g : Y → Z` with `g` surjective between `t_2`, compact topological spaces,
there exists a continuous lift `h : X → Y`, such that `f = g ∘ h`. -/
def CompactT2.Projective : Prop 
goal: Prop

full_name: StoneCech.projective
code: theorem StoneCech.projective [DiscreteTopology X] : CompactT2.Projective (StoneCech X) 
goal: CompactT2.Projective (StoneCech X)

full_name: CompactT2.Projective.extremallyDisconnected
code: protected theorem CompactT2.Projective.extremallyDisconnected [CompactSpace X] [T2Space X]
    (h : CompactT2.Projective X) : ExtremallyDisconnected X 
goal: ExtremallyDisconnected X

full_name: exists_compact_surjective_zorn_subset
code: /-- Lemma 2.4 in [Gleason, *Projective topological spaces*][gleason1958]:
a continuous surjection $\pi$ from a compact space $D$ to a Fréchet space $A$ restricts to
a compact subset $E$ of $D$, such that $\pi$ maps $E$ onto $A$ and satisfies the
"Zorn subset condition", where $\pi(E_0) \ne A$ for any proper closed subset $E_0 \subsetneq E$. -/
lemma exists_compact_surjective_zorn_subset [T1Space A] [CompactSpace D] {π : D → A}
    (π_cont : Continuous π) (π_surj : π.Surjective) : ∃ E : Set D, CompactSpace E ∧ π '' E = univ ∧
    ∀ E₀ : Set E, E₀ ≠ univ → IsClosed E₀ → E.restrict π '' E₀ ≠ univ 
goal: ∃ E, CompactSpace ↑E ∧ π '' E = Set.univ ∧ ∀ (E₀ : Set ↑E), E₀ ≠ Set.univ → IsClosed E₀ → E.restrict π '' E₀ ≠ Set.univ

full_name: image_subset_closure_compl_image_compl_of_isOpen
code: /-- Lemma 2.1 in [Gleason, *Projective topological spaces*][gleason1958]:
if $\rho$ is a continuous surjection from a topological space $E$ to a topological space $A$
satisfying the "Zorn subset condition", then $\rho(G)$ is contained in
the closure of $A \setminus \rho(E \setminus G)$ for any open set $G$ of $E$. -/
lemma image_subset_closure_compl_image_compl_of_isOpen {ρ : E → A} (ρ_cont : Continuous ρ)
    (ρ_surj : ρ.Surjective) (zorn_subset : ∀ E₀ : Set E, E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ)
    {G : Set E} (hG : IsOpen G) : ρ '' G ⊆ closure ((ρ '' Gᶜ)ᶜ) 
goal: ρ '' G ⊆ closure (ρ '' Gᶜ)ᶜ

full_name: ExtremallyDisconnected.disjoint_closure_of_disjoint_isOpen
code: /-- Lemma 2.2 in [Gleason, *Projective topological spaces*][gleason1958]:
in an extremally disconnected space, if $U_1$ and $U_2$ are disjoint open sets,
then $\overline{U_1}$ and $\overline{U_2}$ are also disjoint. -/
lemma ExtremallyDisconnected.disjoint_closure_of_disjoint_isOpen [ExtremallyDisconnected A]
    {U₁ U₂ : Set A} (h : Disjoint U₁ U₂) (hU₁ : IsOpen U₁) (hU₂ : IsOpen U₂) :
    Disjoint (closure U₁) (closure U₂) 
goal: Disjoint (closure U₁) (closure U₂)

full_name: _private.Mathlib.Topology.ExtremallyDisconnected.0.ExtremallyDisconnected.homeoCompactToT2_injective
code: private lemma ExtremallyDisconnected.homeoCompactToT2_injective [ExtremallyDisconnected A]
    [T2Space A] [T2Space E] [CompactSpace E] {ρ : E → A} (ρ_cont : Continuous ρ)
    (ρ_surj : ρ.Surjective) (zorn_subset : ∀ E₀ : Set E, E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ) :
    ρ.Injective 
goal: Function.Injective ρ

full_name: ExtremallyDisconnected.homeoCompactToT2
code: /-- Lemma 2.3 in [Gleason, *Projective topological spaces*][gleason1958]:
a continuous surjection from a compact Hausdorff space to an extremally disconnected Hausdorff space
satisfying the "Zorn subset condition" is a homeomorphism. -/
noncomputable def ExtremallyDisconnected.homeoCompactToT2 [ExtremallyDisconnected A] [T2Space A]
    [T2Space E] [CompactSpace E] {ρ : E → A} (ρ_cont : Continuous ρ) (ρ_surj : ρ.Surjective)
    (zorn_subset : ∀ E₀ : Set E, E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ) : E ≃ₜ A 
goal: E ≃ₜ A

full_name: CompactT2.ExtremallyDisconnected.projective
code: /-- Theorem 2.5 in [Gleason, *Projective topological spaces*][gleason1958]:
in the category of compact spaces and continuous maps,
the projective spaces are precisely the extremally disconnected spaces. -/
protected theorem CompactT2.ExtremallyDisconnected.projective [ExtremallyDisconnected A]
    [CompactSpace A] [T2Space A] : CompactT2.Projective A 
goal: CompactT2.Projective A

full_name: CompactT2.projective_iff_extremallyDisconnected
code: protected theorem CompactT2.projective_iff_extremallyDisconnected [CompactSpace A] [T2Space A] :
    Projective A ↔ ExtremallyDisconnected A 
goal: CompactT2.Projective A ↔ ExtremallyDisconnected A

full_name: instExtremallyDisconnected
code: /-- The sigma-type of extremally disconnected spaces is extremally disconnected. -/
instance instExtremallyDisconnected {ι : Type*} {π : ι → Type*} [∀ i, TopologicalSpace (π i)]
    [h₀ : ∀ i, ExtremallyDisconnected (π i)] : ExtremallyDisconnected (Σ i, π i) 
goal: ExtremallyDisconnected ((i : ι) × π i)