Theorems · Theorem · general topology
Set.Finite.isClosed_biUnion
∀ {X : Type u} {α : Type u_1} [inst : TopologicalSpace X] {s : Set α} {f : α → Set X},
s.Finite → (∀ i ∈ s, IsClosed (f i)) → IsClosed (⋃ i ∈ s, f i)- Defined in
- Mathlib.Topology.Basic
- 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.
Cites9
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.iUnionstatement and proof · cited by 2,483
- IsOpenproof · cited by 2,400
- Set.Finitestatement and proof · cited by 1,814
- IsClosedstatement and proof · cited by 1,639
- Set.iInterproof · cited by 1,084
- Set.compl_iUnionproof · cited by 32
- Set.Finite.isOpen_biInterproof · cited by 10
Cited by11
Results whose statement or proof uses this declaration.
- Set.Finite.isClosedproof · cited by 6
- isClosed_biUnion_finsetproof · cited by 3
- closure_sUnion_irreducibleComponents_sdiff_singletonproof · cited by 2
- Set.Finite.isClopen_biUnionproof · cited by 2
- PrimeSpectrum.isOpen_singleton_tfae_of_isNoetherian_of_isJacobsonRingproof · cited by 2
- IsClosed.relImage_of_finiteproof · cited by 1
- Topology.IsLower.isClosed_upperClosureproof · cited by 1
- MeasureTheory.isTightMeasureSet_of_isCompact_closureproof · cited by 0
- AlgebraicGeometry.Scheme.Hom.dense_smoothLocus_of_perfectFieldproof · cited by 0
- JacobsonSpace.discreteTopologyproof · cited by 0