Theorems · Theorem · general topology
Set.Finite.isOpen_sInter
∀ {X : Type u} [inst : TopologicalSpace X] {s : Set (Set X)}, s.Finite → (∀ t ∈ s, IsOpen t) → IsOpen (⋂₀ s)- Defined in
- Mathlib.Topology.Basic
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 68 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.
Cites10
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.sInterstatement and proof · cited by 225
- isOpen_univproof · cited by 112
- IsOpen.interproof · cited by 98
- Set.Finite.induction_onproof · cited by 39
- Set.sInter_emptyproof · cited by 16
- Set.sInter_insertproof · cited by 8
Cited by6
Results whose statement or proof uses this declaration.
- Set.Finite.isOpen_biInterproof · cited by 10
- TopologicalSpace.isTopologicalBasis_of_subbasisproof · cited by 6
- isOpen_iInter_of_finiteproof · cited by 6
- TopologicalSpace.vietoris.isTopologicalBasisproof · cited by 2
- isIrreducible_iff_sInterproof · cited by 1
- Set.Finite.dense_sInterproof · cited by 1