Theorems · Theorem · general topology
Finset.isCompact_biUnion
∀ {X : Type u} {ι : Type u_1} [inst : TopologicalSpace X] (s : Finset ι) {f : ι → Set X},
(∀ i ∈ s, IsCompact (f i)) → IsCompact (⋃ i ∈ s, f i)- Defined in
- Mathlib.Topology.Compactness.Compact
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 82 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.
Cites7
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.iUnionstatement · cited by 2,483
- IsCompactstatement and proof · cited by 1,282
- Finset.finite_toSetproof · cited by 210
- Set.Finite.isCompact_biUnionproof · cited by 11
Cited by5
Results whose statement or proof uses this declaration.
- exists_compact_supersetproof · cited by 6
- AlgebraicGeometry.isAffine_of_isAffineOpen_basicOpenproof · cited by 2
- exists_mem_nhdsSet_isCompact_mapsToproof · cited by 2
- AlgebraicGeometry.compactSpace_of_universallyClosedproof · cited by 1
- QuasiSeparatedSpace.of_isOpenCoverproof · cited by 1