Theorems · Theorem · general topology
IsPathConnected.isConnected
∀ {X : Type u_1} [inst : TopologicalSpace X] {F : Set X}, IsPathConnected F → IsConnected FA path-connected set is connected.
(See Counterexamples.TopologistsSineCurve for the standard counterexample showing that the
converse is false.)
- Defined in
- Mathlib.Topology.Connected.PathConnected
- Cited by
- 7 results in Mathlib
- Foundations
- Depth 132 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.
Cites5
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
- IsConnectedstatement · cited by 116
- IsPathConnectedstatement and proof · cited by 65
- isConnected_iff_connectedSpaceproof · cited by 7
Cited by7
Results whose statement or proof uses this declaration.
- Convex.isConnectedproof · cited by 3
- isConnected_compl_singleton_of_one_lt_rankproof · cited by 2
- isConnected_sphereproof · cited by 2
- Set.Countable.isConnected_compl_of_one_lt_rankproof · cited by 1
- Metric.isConnected_eballproof · cited by 1
- isConnected_compl_of_one_lt_codimproof · cited by 1
- Metric.isConnected_ballproof · cited by 0