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[Merged by Bors] - chore(Topology/Compactness/CountablyCompact): generalize theorem #39600

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[Merged by Bors] - chore(Topology/Compactness/CountablyCompact): generalize theorem #39600

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4f9adc1
free
plp127 May 19, 2026
e3e7ca0
add deprecation
plp127 May 19, 2026
f8c701e
add deprecation :(((
plp127 May 23, 2026
c640c96
unadd deprecation :)
plp127 Jun 3, 2026
7b9b73e
Merge branch 'master' into aliu/inducing-free
plp127 Jun 3, 2026
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9 changes: 6 additions & 3 deletions Mathlib/Topology/Compactness/CountablyCompact.lean
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Expand Up @@ -261,9 +261,9 @@ instance (priority := 50) [SequentialSpace E] [CountablyCompactSpace E] :
· simp only [image_eq_iUnion, mem_Ici, iUnion_ge_eq_iUnion_nat_add _ (k + 1)]
exact iUnion_mono fun i => subset_closure

/-- If `f : X → Y` is an embedding map, the image `f '' s` of a set `s` is sequentially compact
/-- If `f : X → Y` is an inducing map, the image `f '' s` of a set `s` is sequentially compact
if and only if `s` is sequentially compact. -/
theorem Topology.IsEmbedding.isSeqCompact_iff {f : E → F} (hf : IsEmbedding f) :
theorem Topology.IsInducing.isSeqCompact_iff {f : E → F} (hf : IsInducing f) :
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plp127 marked this conversation as resolved.
IsSeqCompact A ↔ IsSeqCompact (f '' A) where
mp hA x hx := by
choose y hy using hx
Expand Down Expand Up @@ -358,10 +358,13 @@ theorem IsLindelof.isCompact (hA : IsCountablyCompact A) (hl : IsLindelof A) :
· exact ⟨∅, by simp_all⟩

/-- A countably compact Lindelöf space is compact. -/
theorem LindelofSpace.CompactSpace {X : Type*} [TopologicalSpace X]
theorem LindelofSpace.compactSpace {X : Type*} [TopologicalSpace X]
[LindelofSpace X] [h : CountablyCompactSpace X] : CompactSpace X where
isCompact_univ := isLindelof_univ.isCompact h.isCountablyCompact_univ

@[deprecated (since := "2026-05-19")]
alias LindelofSpace.CompactSpace := LindelofSpace.compactSpace

/-- In a Hereditarily Lindelöf space, a countably compact set is compact. -/
theorem IsCountablyCompact.isCompact [HereditarilyLindelofSpace E]
(hA : IsCountablyCompact A) : IsCompact A :=
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