Theorems · Theorem · general topology
isOpen_iInter_of_finite
∀ {X : Type u} {ι : Sort v} [inst : TopologicalSpace X] [Finite ι] {s : ι → Set X},
(∀ (i : ι), IsOpen (s i)) → IsOpen (⋂ i, s i)- Defined in
- Mathlib.Topology.Basic
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 80 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
- IsOpenstatement and proof · cited by 2,400
- Set.iInterstatement · cited by 1,084
- Set.forall_mem_rangeproof · cited by 135
- Set.finite_rangeproof · cited by 58
- Set.Finite.isOpen_sInterproof · cited by 6
Cited by6
Results whose statement or proof uses this declaration.
- isClosed_iUnion_of_finiteproof · cited by 6
- IsClosedMap.isEvenlyCovered_of_openPartialHomeomorphproof · cited by 2
- TopologicalSpace.vietoris.continuous_range_of_finiteproof · cited by 1
- ProfiniteGrp.denseRange_toLimitproof · cited by 1
- isClopen_iInter_of_finiteproof · cited by 0
- isOpen.dynEntourageproof · cited by 0