Theorems · Theorem · general topology
isPathConnected_pathComponentIn
∀ {X : Type u_1} [inst : TopologicalSpace X] {x : X} {F : Set X}, x ∈ F → IsPathConnected (pathComponentIn F x)- Defined in
- Mathlib.Topology.Connected.PathConnected
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 126 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.
Cites14
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Setstatement and proof · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- Set.Elemproof · cited by 7,166
- unitIntervalproof · cited by 607
- Pathproof · cited by 318
- IsPathConnectedstatement · cited by 65
- JoinedInproof · cited by 40
- Path.castproof · cited by 22
- pathComponentInstatement and proof · cited by 19
- mem_pathComponentIn_selfproof · cited by 7
- Path.extend_extends'proof · cited by 3
Cited by6
Results whose statement or proof uses this declaration.
- locallyPathConnectedSpace_iff_isOpen_pathComponentInproof · cited by 3
- LocallyPathConnectedSpace.coinducedproof · cited by 2
- isPathConnected_pathComponentproof · cited by 1
- ChartedSpace.locallyPathConnectedSpaceproof · cited by 1
- Pi.locallyPathConnectedSpace_of_finite_not_pathConnectedSpaceproof · cited by 1
- isOpen_isPathConnected_basisproof · cited by 0