Theorems · Definition · general topology
pathComponentIn
{X : Type u_1} → [TopologicalSpace X] → Set X → X → Set XThe path component of x in F is the set of points that can be joined to x in F.
- Defined in
- Mathlib.Topology.Connected.PathConnected
- Cited by
- 19 results in Mathlib
- Foundations
- Depth 120 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.
Cites4
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.ofPredproof · cited by 6,101
- JoinedInproof · cited by 40
Cited by19
Results whose statement or proof uses this declaration.
- mem_pathComponentIn_selfstatement · cited by 7
- pathComponentIn_subsetstatement and proof · cited by 7
- isPathConnected_pathComponentInstatement and proof · cited by 6
- IsOpen.pathComponentInstatement and proof · cited by 5
- pathComponentIn_congrstatement and proof · cited by 3
- locallyPathConnectedSpace_iff_isOpen_pathComponentInstatement and proof · cited by 3
- pathComponentIn_univstatement · cited by 3
- IsPathConnected.subset_pathComponentInstatement · cited by 2
- pathComponentIn_mem_nhdsstatement and proof · cited by 2
- LocallyPathConnectedSpace.coinducedproof · cited by 2
- Pi.locallyPathConnectedSpace_of_finite_not_pathConnectedSpaceproof · cited by 1
- locallyPathConnectedSpace_iff_pathComponentIn_mem_nhdsstatement and proof · cited by 1