Theorems · Definition · general topology
IsTotallyDisconnected
{α : Type u} → [TopologicalSpace α] → Set α → PropA set s is called totally disconnected if every subset t ⊆ s which is preconnected is
a subsingleton, i.e. either empty or a singleton.
- Cited by
- 14 results in Mathlib
- Foundations
- Depth 7 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
- Set.Subsingletonproof · cited by 276
- IsPreconnectedproof · cited by 205
Cited by16
Results whose statement or proof uses this declaration.
- TotallyDisconnectedSpace.isTotallyDisconnected_univstatement · cited by 3
- totallyDisconnectedSpace_iffstatement and proof · cited by 2
- Topology.IsEmbedding.isTotallyDisconnected_rangestatement and proof · cited by 2
- TotallyDisconnectedSpace.casesOnstatement and proof · cited by 1
- Topology.IsEmbedding.isTotallyDisconnectedstatement and proof · cited by 1
- Topology.IsEmbedding.isTotallyDisconnected_imagestatement and proof · cited by 1
- totallyDisconnectedSpace_subtype_iffstatement and proof · cited by 1
- isTotallyDisconnected_of_imagestatement and proof · cited by 1
- isTotallyDisconnected_of_isTotallySeparatedstatement · cited by 1
- Set.Countable.isTotallyDisconnectedstatement · cited by 0
- IsTotallySeparated.isTotallyDisconnectedstatement · cited by 0
- TotallyDisconnectedSpace.recOnstatement and proof · cited by 0