Theorems · Theorem · general topology
Set.Finite.isCompact_biUnion
∀ {X : Type u} {ι : Type u_1} [inst : TopologicalSpace X] {s : Set ι} {f : ι → Set X},
s.Finite → (∀ i ∈ s, IsCompact (f i)) → IsCompact (⋃ i ∈ s, f i)- Defined in
- Mathlib.Topology.Compactness.Compact
- Cited by
- 11 results in Mathlib
- Foundations
- Depth 81 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.
Cites13
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
- nhdsproof · cited by 5,554
- Set.iUnionstatement and proof · cited by 2,483
- Set.Finitestatement and proof · cited by 1,814
- IsCompactstatement and proof · cited by 1,282
- Ultrafilterproof · cited by 193
- Ultrafilter.toFilterproof · cited by 172
- Filter.le_principal_iffproof · cited by 87
- Set.mem_iUnion₂proof · cited by 70
- isCompact_iff_ultrafilter_le_nhds'proof · cited by 6
- IsCompact.ultrafilter_le_nhdsproof · cited by 5
Cited by11
Results whose statement or proof uses this declaration.
- Set.Finite.isCompactproof · cited by 10
- Finset.isCompact_biUnionproof · cited by 5
- isCompact_open_iff_eq_finite_iUnion_of_isTopologicalBasisproof · cited by 4
- isCompact_accumulateproof · cited by 2
- Set.Finite.isCompact_sUnionproof · cited by 1
- isCompact_closure_of_isTightMeasureSetproof · cited by 1
- Set.isCompact_sigmaproof · cited by 1
- IsOpenMap.exists_opens_image_eq_of_prespectralSpaceproof · cited by 1
- QuasiSeparatedSpace.of_isTopologicalBasisproof · cited by 0
- AlgebraicGeometry.quasiCompact_iff_forall_isAffineOpenproof · cited by 0
- Filter.HasBasis.nhds_continuousMapConstproof · cited by 0