Theorems · Theorem · general topology
isCountablyCompact_iff_countable_open_cover
∀ {E : Type u_2} [inst : TopologicalSpace E] {A : Set E},
IsCountablyCompact A ↔ ∀ (U : ℕ → Set E), (∀ (i : ℕ), IsOpen (U i)) → A ⊆ ⋃ i, U i → ∃ t, A ⊆ ⋃ i ∈ t, U iA set is countably compact if and only if every countable open cover has a finite subcover.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 91 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites32
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- Finsetstatement and proof · cited by 13,712
- Set.imageproof · cited by 5,609
- LE.le.transproof · cited by 3,151
- Filter.Eventuallyproof · cited by 3,134
- Compl.complproof · cited by 2,925
- Set.iUnionstatement and proof · cited by 2,483
- Filter.atTopproof · cited by 2,405
- IsOpenstatement and proof · cited by 2,400
- le_rflproof · cited by 1,558
- Monotoneproof · cited by 1,397
Cited by1
Results whose statement or proof uses this declaration.
- IsLindelof.isCompactproof · cited by 2