Theorems · Theorem · general topology
isOpen_singleton_iff_nhds_eq_pure
∀ {X : Type u} [inst : TopologicalSpace X] (x : X), IsOpen {x} ↔ nhds x = pure x- Defined in
- Mathlib.Topology.Neighborhoods
- Cited by
- 9 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.
Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- Filterstatement · cited by 8,121
- nhdsstatement and proof · cited by 5,554
- IsOpenstatement · cited by 2,400
- LE.le.ge_iff_eq'proof · cited by 50
- pure_le_nhdsproof · cited by 39
Cited by9
Results whose statement or proof uses this declaration.
- SuccOrder.nhds_eq_pureproof · cited by 3
- TopologicalSpace.isClosed_range_singletonproof · cited by 2
- PredOrder.nhds_eq_pureproof · cited by 2
- nhds_nhdsAdjoint_of_neproof · cited by 2
- Subgroup.discreteTopology_iff_of_finiteIndexproof · cited by 1
- isOpen_singleton_iff_punctured_nhdsproof · cited by 1
- AddSubgroup.discreteTopology_iff_of_finiteIndexproof · cited by 1
- le_nhdsAdjoint_iffproof · cited by 0
- ENat.isOpen_singletonproof · cited by 0