Theorems · Theorem · general topology
isClosed_singleton
∀ {X : Type u_1} [inst : TopologicalSpace X] [T1Space X] {x : X}, IsClosed {x}- Defined in
- Mathlib.Topology.Separation.Basic
- Cited by
- 66 results in Mathlib
- Foundations
- Depth 5 from the axioms · uses no axioms
- Assumes
- TopologicalSpaceT1Space
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
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
- IsClosedstatement · cited by 1,639
- T1Spacestatement and proof · cited by 275
- T1Space.t1proof · cited by 11
Cited by66
Results whose statement or proof uses this declaration.
- nhdsWithin_insert_of_neproof · cited by 9
- closure_singletonproof · cited by 8
- ContinuousLinearMap.isClosed_kerproof · cited by 6
- Set.Finite.isClosedproof · cited by 6
- IsOpen.exists_is_const_of_fderiv_eq_zeroproof · cited by 3
- Set.Subsingleton.isClosedproof · cited by 3
- Matrix.SpecialLinearGroup.isClosedEmbedding_valproof · cited by 2
- IsPreconnected.preperfect_of_nontrivialproof · cited by 2
- ContinuousAlternatingMap.isClosed_range_toContinuousMultilinearMapproof · cited by 2
- Units.isClosedEmbedding_embedProductproof · cited by 2