Theorems · Definition · general topology
IsLocallyClosed
{X : Type u} → [TopologicalSpace X] → Set X → PropA set is locally closed if it is the intersection of some open set and some closed set.
Also see isLocallyClosed_tfae and other lemmas in Mathlib/Topology/LocallyClosed.lean.
- Defined in
- Mathlib.Topology.Defs.Basic
- Cited by
- 44 results in Mathlib
- Foundations
- Depth 6 from the axioms · uses no axioms
- Assumes
- TopologicalSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites4
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
- IsOpenproof · cited by 2,400
- IsClosedproof · cited by 1,639
Cited by46
Results whose statement or proof uses this declaration.
- IsOpen.isLocallyClosedstatement · cited by 20
- IsClosed.isLocallyClosedstatement · cited by 12
- IsLocallyClosed.interstatement and proof · cited by 6
- MeasureTheory.locallyIntegrableOn_iffstatement and proof · cited by 5
- nonempty_inter_closedPointsstatement and proof · cited by 5
- IsLocallyClosed.imagestatement and proof · cited by 5
- Topology.IsInducing.locallyCompactSpacestatement and proof · cited by 4
- jacobsonSpace_iff_locallyClosedstatement and proof · cited by 4
- isClosed_singleton_of_isLocallyClosed_singletonstatement and proof · cited by 3
- JacobsonSpace.of_isOpenEmbeddingproof · cited by 3
- Topology.IsInducing.isLocallyClosed_iffstatement · cited by 2
- JacobsonSpace.closure_inter_closedPoints_eq_closurestatement and proof · cited by 2