Theorems · Theorem · general topology
pathComponent_eq_connectedComponent
∀ {X : Type u_1} [inst : TopologicalSpace X] [LocallyPathConnectedSpace X] (x : X),
pathComponent x = connectedComponent x- Cited by
- 2 results in Mathlib
- Foundations
- Depth 132 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- LE.le.antisymmproof · cited by 507
- connectedComponentstatement · cited by 68
- LocallyPathConnectedSpacestatement and proof · cited by 53
- pathComponentstatement · cited by 30
- IsClopen.connectedComponent_subsetproof · cited by 5
- mem_pathComponent_selfproof · cited by 4
- IsClopen.pathComponentproof · cited by 2
- pathComponent_subset_componentproof · cited by 2
Cited by2
Results whose statement or proof uses this declaration.
- connectedComponent_eq_iff_joinedproof · cited by 1
- Unitary.mem_pathComponentOne_iffproof · cited by 0