Theorems · Theorem · general topology
isClosed_iInter
∀ {X : Type u} {ι : Sort v} [inst : TopologicalSpace X] {f : ι → Set X},
(∀ (i : ι), IsClosed (f i)) → IsClosed (⋂ i, f i)- Defined in
- Mathlib.Topology.Basic
- Cited by
- 43 results in Mathlib
- Foundations
- Depth 65 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.
Cites6
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
- IsClosedstatement and proof · cited by 1,639
- Set.iInterstatement · cited by 1,084
- Set.forall_mem_rangeproof · cited by 135
- isClosed_sInterproof · cited by 10
Cited by43
Results whose statement or proof uses this declaration.
- isClosed_biInterproof · cited by 16
- Submodule.isClosed_orthogonalproof · cited by 4
- isClosed_setOfPred_blockTriangularproof · cited by 2
- isClosed_setOfPred_convexOnproof · cited by 2
- ContinuousAlternatingMap.isClosed_range_toContinuousMultilinearMapproof · cited by 2
- connectedComponent_eq_iInter_isClopenproof · cited by 2
- CantorBendixson.isClosed_iteratedDerivedSetproof · cited by 2
- ContinuousMultilinearMap.isLeast_opNormproof · cited by 2
- CommRingCat.HomTopology.isClosedEmbedding_precomp_of_surjectiveproof · cited by 1
- isClosed_setOfPred_map_addproof · cited by 1
- isClosed_setOfPred_map_invproof · cited by 1
- isClosed_setOfPred_map_mulproof · cited by 1