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Theorems · Theorem · general topology

isOpen_sInter

∀ {α : Type u_3} [inst : TopologicalSpace α] [AlexandrovDiscrete α] {S : Set (Set α)},
  (∀ s ∈ S, IsOpen s) → IsOpen (⋂₀ S)
Defined in
Mathlib.Topology.AlexandrovDiscrete
Cited by
5 results in Mathlib
Foundations
Depth 7 from the axioms · uses no axioms
Assumes
TopologicalSpaceAlexandrovDiscrete
Mathlib docsSourceLoogleLeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

isOpen_nhdsKer · cited by 4isOpen_nhdsKerisOpen_iInter · cited by 3isOpen_iInteralexandrovDiscrete_iSup · cited by 0alexandrovDiscrete_iSupAlexandrovDiscrete.sup · cited by 0AlexandrovDiscrete.supisClopen_sInter · cited by 0isClopen_sInterSet · cited by 53352SetTopologicalSpace · cited by 24529TopologicalSpaceIsOpen · cited by 2400IsOpenSet.sInter · cited by 225Set.sInterAlexandrovDiscrete · cited by 45AlexandrovDiscreteAlexandrovDiscrete.isOpen_sInter · cited by 1AlexandrovDiscrete.isOpen…isOpen_sInterCITED BYCITES

Cites6

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Cited by5

Results whose statement or proof uses this declaration.