Theorems · Theorem · general topology
isCompact_iUnion
∀ {X : Type u} [inst : TopologicalSpace X] {ι : Sort u_2} {f : ι → Set X} [Finite ι],
(∀ (i : ι), IsCompact (f i)) → IsCompact (⋃ i, f i)- Defined in
- Mathlib.Topology.Compactness.Compact
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 83 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpaceFinite
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
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
- Finitestatement and proof · cited by 3,029
- Set.iUnionstatement · cited by 2,483
- IsCompactstatement and proof · cited by 1,282
- Set.forall_mem_rangeproof · cited by 135
- Set.finite_rangeproof · cited by 58
- Set.Finite.isCompact_sUnionproof · cited by 1
Cited by6
Results whose statement or proof uses this declaration.
- IsCompact.unionproof · cited by 23
- IsCompactOpenCovered.of_iUnion_eq_of_finiteproof · cited by 2
- AlgebraicGeometry.exists_preimage_eqproof · cited by 1
- TopologicalSpace.NoetherianSpace.iUnionproof · cited by 0
- AlgebraicGeometry.Scheme.OpenCover.compactSpaceproof · cited by 0
- Polynomial.isCompact_image_comap_of_monicproof · cited by 0