Theorems · Theorem · general topology
isPreconnected_connectedComponent
∀ {α : Type u} [inst : TopologicalSpace α] {x : α}, IsPreconnected (connectedComponent x)- Defined in
- Mathlib.Topology.Connected.Basic
- Cited by
- 12 results in Mathlib
- Foundations
- Depth 62 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.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setproof · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- Set.ofPredproof · cited by 6,101
- IsPreconnectedstatement and proof · cited by 205
- connectedComponentstatement · cited by 68
- isPreconnected_sUnionproof · cited by 3
Cited by12
Results whose statement or proof uses this declaration.
- isConnected_connectedComponentproof · cited by 7
- IsClopen.connectedComponent_subsetproof · cited by 5
- isPreconnected_connectedComponentInproof · cited by 4
- connectedComponentIn_univproof · cited by 3
- totallyDisconnectedSpace_iff_connectedComponent_subsingletonproof · cited by 1
- preconnectedSpace_iff_connectedComponentproof · cited by 1
- Continuous.image_connectedComponent_eq_singletonproof · cited by 1
- connectedComponent_prodproof · cited by 0
- Pi.locallyConnectedSpace_iffproof · cited by 0
- connectedSpace_iff_connectedComponentproof · cited by 0
- IsClopen.connectedComponentIn_eqproof · cited by 0
- connectedComponent_piproof · cited by 0