Theorems · Theorem · general topology
Set.Finite.isOpen_biInter
∀ {X : Type u} {α : Type u_1} [inst : TopologicalSpace X] {s : Set α} {f : α → Set X},
s.Finite → (∀ i ∈ s, IsOpen (f i)) → IsOpen (⋂ i ∈ s, f i)- Defined in
- Mathlib.Topology.Basic
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 80 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
- IsOpenstatement and proof · cited by 2,400
- Set.Finitestatement and proof · cited by 1,814
- Set.iInterstatement · cited by 1,084
- Set.Finite.imageproof · cited by 96
- Set.forall_mem_imageproof · cited by 65
- Set.sInter_imageproof · cited by 26
- Set.Finite.isOpen_sInterproof · cited by 6
Cited by10
Results whose statement or proof uses this declaration.
- Set.Finite.isClosed_biUnionproof · cited by 11
- isOpen_set_piproof · cited by 8
- isOpen_biInter_finsetproof · cited by 7
- TopologicalSpace.vietoris.isTopologicalBasisproof · cited by 2
- TopologicalSpace.IsTopologicalBasis.compactsproof · cited by 2
- TopologicalSpace.exists_countable_of_generateFromproof · cited by 1
- TopologicalSpace.IsTopologicalBasis.vietorisproof · cited by 1
- TopologicalSpace.IsTopologicalBasis.nonemptyCompactsproof · cited by 1
- Set.Finite.isClopen_biInterproof · cited by 1
- isTopologicalBasis_biInter_Ioi_Iio_of_generateFromproof · cited by 0