Theorems · Theorem · general topology
UniformSpace.hausdorff.isOpen_inter_nonempty_of_isOpen
∀ {α : Type u_1} [inst : UniformSpace α] {U : Set α}, IsOpen U → IsOpen {s | (s ∩ U).Nonempty}- Defined in
- Mathlib.Topology.UniformSpace.Closeds
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 69 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- UniformSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites16
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
- Set.ofPredstatement and proof · cited by 6,101
- nhdsproof · cited by 5,554
- Set.Nonemptystatement and proof · cited by 2,627
- IsOpenstatement and proof · cited by 2,400
- UniformSpacestatement and proof · cited by 2,040
- uniformityproof · cited by 765
- UniformSpace.ballproof · cited by 113
- SetRel.preimageproof · cited by 56
- SetRel.imageproof · cited by 49
- isOpen_iff_mem_nhdsproof · cited by 48
- hausdorffEntourageproof · cited by 35
Cited by4
Results whose statement or proof uses this declaration.
- IsClosed.powerset_hausdorffproof · cited by 2
- UniformSpace.hausdorff.isClosedEmbedding_singletonproof · cited by 1
- IsCompact.nhds_hausdorff_eq_nhds_vietorisproof · cited by 0
- TopologicalSpace.Closeds.isOpen_inter_nonempty_of_isOpenproof · cited by 0