Theorems · Theorem · general topology
pathComponent_subset_component
∀ {X : Type u_1} [inst : TopologicalSpace X] (x : X), pathComponent x ⊆ connectedComponent x- Defined in
- Mathlib.Topology.Connected.PathConnected
- Cited by
- 2 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.
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
- connectedComponentstatement · cited by 68
- pathComponentstatement and proof · cited by 30
- Path.sourceproof · cited by 26
- Path.targetproof · cited by 24
- Joined.somePathproof · cited by 12
- IsConnected.subset_connectedComponentproof · cited by 6
- Path.continuousproof · cited by 3
- isConnected_rangeproof · cited by 3
Cited by2
Results whose statement or proof uses this declaration.
- pathComponent_eq_connectedComponentproof · cited by 2
- biUnion_connectedComponent_pathComponent_eqproof · cited by 0