Theorems · Theorem · general topology
IsOpen.isLocallyClosed
∀ {X : Type u} {s : Set X} [inst : TopologicalSpace X], IsOpen s → IsLocallyClosed s- Defined in
- Mathlib.Topology.Basic
- Cited by
- 20 results in Mathlib
- Foundations
- Depth 66 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.
Cites7
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
- Set.univproof · cited by 3,945
- IsOpenstatement and proof · cited by 2,400
- Set.inter_univproof · cited by 198
- IsLocallyClosedstatement · cited by 44
- isClosed_univproof · cited by 43
Cited by20
Results whose statement or proof uses this declaration.
- jacobsonSpace_iff_locallyClosedproof · cited by 4
- Topology.IsOpenEmbedding.locallyCompactSpaceproof · cited by 3
- JacobsonSpace.of_isOpenEmbeddingproof · cited by 3
- isLocallyClosed_tfaeproof · cited by 2
- mellin_convergent_of_isBigO_scalarproof · cited by 2
- mellin_hasDerivAt_of_isBigO_rpowproof · cited by 2
- JacobsonSpace.closure_inter_closedPoints_eq_closureproof · cited by 2
- AlgebraicGeometry.Scheme.Hom.QuasiFiniteAt.isClopen_singleton_asFiberproof · cited by 1
- TopologicalSpace.IsOpenCover.jacobsonSpace_iffproof · cited by 1
- MeasurableSet.residualEq_isOpenproof · cited by 1
- isLocallyClosed_Iioproof · cited by 1