Theorems · Theorem · general topology
isPreconnected_iff_preconnectedSpace
∀ {α : Type u} [inst : TopologicalSpace α] {s : Set α}, IsPreconnected s ↔ PreconnectedSpace ↑s- Defined in
- Mathlib.Topology.Connected.Basic
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 80 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.
Cites12
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.Elemstatement and proof · cited by 7,166
- Set.image_univproof · cited by 322
- Continuous.continuousOnproof · cited by 311
- IsPreconnectedstatement and proof · cited by 205
- Subtype.range_coe_subtypeproof · cited by 170
- continuous_subtype_valproof · cited by 159
- PreconnectedSpacestatement and proof · cited by 64
- PreconnectedSpace.isPreconnected_univproof · cited by 26
- IsPreconnected.imageproof · cited by 24
- Subtype.preconnectedSpaceproof · cited by 4
Cited by4
Results whose statement or proof uses this declaration.
- isConnected_iff_connectedSpaceproof · cited by 7
- MeromorphicOn.exists_meromorphicOrderAt_ne_top_iff_forallproof · cited by 7
- IsPreconnected.constant_of_mapsToproof · cited by 1
- IsPreconnected.isDiscrete_iff_subsingletonproof · cited by 0