Theorems · Theorem · general topology
isPreconnected_connectedComponentIn
∀ {α : Type u} [inst : TopologicalSpace α] {x : α} {F : Set α}, IsPreconnected (connectedComponentIn F x)- Defined in
- Mathlib.Topology.Connected.Basic
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 79 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.
Cites8
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
- IsPreconnectedstatement and proof · cited by 205
- Topology.IsInducing.subtypeValproof · cited by 43
- connectedComponentInstatement · cited by 33
- isPreconnected_emptyproof · cited by 14
- isPreconnected_connectedComponentproof · cited by 12
- Topology.IsInducing.isPreconnected_imageproof · cited by 11
Cited by4
Results whose statement or proof uses this declaration.
- connectedComponentIn_univproof · cited by 3
- ContinuousOn.image_connectedComponentIn_subsetproof · cited by 3
- Pi.locallyConnectedSpace_of_finite_not_preconnectedSpaceproof · cited by 1
- IsClopen.connectedComponentIn_eqproof · cited by 0