Theorems · Theorem · general topology
isSigmaCompact_biUnion
∀ {X : Type u_1} {ι : Type u_3} [inst : TopologicalSpace X] {s : Set ι} {S : ι → Set X},
s.Countable → (∀ i ∈ s, IsSigmaCompact (S i)) → IsSigmaCompact (⋃ i ∈ s, S i)Countable unions of σ-compact sets are σ-compact.
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- Foundations
- Depth 72 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpace
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Cites10
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
- Set.Elemproof · cited by 7,166
- Set.iUnionstatement · cited by 2,483
- Countableproof · cited by 633
- Set.Countablestatement and proof · cited by 545
- Set.biUnion_eq_iUnionproof · cited by 41
- IsSigmaCompactstatement and proof · cited by 24
- Set.countable_coe_iffproof · cited by 15
- isSigmaCompact_iUnionproof · cited by 2
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