File: Mathlib/Topology/QuasiSeparated.lean

full_name: IsQuasiSeparated
code: /-- A subset `s` of a topological space is quasi-separated if the intersections of any pairs of
compact open subsets of `s` are still compact.

Note that this is equivalent to `s` being a `QuasiSeparatedSpace` only when `s` is open. -/
def IsQuasiSeparated (s : Set α) : Prop 
goal: Prop

full_name: quasiSeparatedSpace_iff
code: /-- A topological space is quasi-separated if the intersections of any pairs of compact open
subsets are still compact. -/
@[mk_iff]
class QuasiSeparatedSpace (α : Type*) [TopologicalSpace α] : Prop where
  /-- The intersection of two open compact subsets of a quasi-separated space is compact. -/
  inter_isCompact :
    ∀ U V : Set α, IsOpen U → IsCompact U → IsOpen V → IsCompact V → IsCompact (U ∩ V)
goal: QuasiSeparatedSpace α ↔ ∀ (U V : Set α), IsOpen U → IsCompact U → IsOpen V → IsCompact V → IsCompact (U ∩ V)

full_name: QuasiSeparatedSpace
code: /-- A topological space is quasi-separated if the intersections of any pairs of compact open
subsets are still compact. -/
@[mk_iff]
class QuasiSeparatedSpace (α : Type*) [TopologicalSpace α] : Prop where
  /-- The intersection of two open compact subsets of a quasi-separated space is compact. -/
  inter_isCompact :
    ∀ U V : Set α, IsOpen U → IsCompact U → IsOpen V → IsCompact V → IsCompact (U ∩ V)
goal: Prop

full_name: QuasiSeparatedSpace.inter_isCompact
code: /-- A topological space is quasi-separated if the intersections of any pairs of compact open
subsets are still compact. -/
@[mk_iff]
class QuasiSeparatedSpace (α : Type*) [TopologicalSpace α] : Prop where
  /-- The intersection of two open compact subsets of a quasi-separated space is compact. -/
  inter_isCompact :
    ∀ U V : Set α, IsOpen U → IsCompact U → IsOpen V → IsCompact V → IsCompact (U ∩ V)
goal: IsOpen U → IsCompact U → IsOpen V → IsCompact V → IsCompact (U ∩ V)

full_name: isQuasiSeparated_univ_iff
code: theorem isQuasiSeparated_univ_iff {α : Type*} [TopologicalSpace α] :
    IsQuasiSeparated (Set.univ : Set α) ↔ QuasiSeparatedSpace α 
goal: IsQuasiSeparated Set.univ ↔ QuasiSeparatedSpace α

full_name: isQuasiSeparated_univ
code: theorem isQuasiSeparated_univ {α : Type*} [TopologicalSpace α] [QuasiSeparatedSpace α] :
    IsQuasiSeparated (Set.univ : Set α) 
goal: IsQuasiSeparated Set.univ

full_name: IsQuasiSeparated.image_of_embedding
code: theorem IsQuasiSeparated.image_of_embedding {s : Set α} (H : IsQuasiSeparated s) (h : Embedding f) :
    IsQuasiSeparated (f '' s) 
goal: IsQuasiSeparated (f '' s)

full_name: OpenEmbedding.isQuasiSeparated_iff
code: theorem OpenEmbedding.isQuasiSeparated_iff (h : OpenEmbedding f) {s : Set α} :
    IsQuasiSeparated s ↔ IsQuasiSeparated (f '' s) 
goal: IsQuasiSeparated s ↔ IsQuasiSeparated (f '' s)

full_name: isQuasiSeparated_iff_quasiSeparatedSpace
code: theorem isQuasiSeparated_iff_quasiSeparatedSpace (s : Set α) (hs : IsOpen s) :
    IsQuasiSeparated s ↔ QuasiSeparatedSpace s 
goal: IsQuasiSeparated s ↔ QuasiSeparatedSpace ↑s

full_name: IsQuasiSeparated.of_subset
code: theorem IsQuasiSeparated.of_subset {s t : Set α} (ht : IsQuasiSeparated t) (h : s ⊆ t) :
    IsQuasiSeparated s 
goal: IsQuasiSeparated s

full_name: T2Space.to_quasiSeparatedSpace
code: instance (priority := 100) T2Space.to_quasiSeparatedSpace [T2Space α] : QuasiSeparatedSpace α 
goal: QuasiSeparatedSpace α

full_name: NoetherianSpace.to_quasiSeparatedSpace
code: instance (priority := 100) NoetherianSpace.to_quasiSeparatedSpace [NoetherianSpace α] :
    QuasiSeparatedSpace α 
goal: QuasiSeparatedSpace α

full_name: IsQuasiSeparated.of_quasiSeparatedSpace
code: theorem IsQuasiSeparated.of_quasiSeparatedSpace (s : Set α) [QuasiSeparatedSpace α] :
    IsQuasiSeparated s 
goal: IsQuasiSeparated s

full_name: QuasiSeparatedSpace.of_openEmbedding
code: theorem QuasiSeparatedSpace.of_openEmbedding (h : OpenEmbedding f) [QuasiSeparatedSpace β] :
    QuasiSeparatedSpace α 
goal: QuasiSeparatedSpace α