Theorems · Theorem · general topology
pathComponentIn_subset
∀ {X : Type u_1} [inst : TopologicalSpace X] {x : X} {F : Set X}, pathComponentIn F x ⊆ F- Defined in
- Mathlib.Topology.Connected.PathConnected
- Cited by
- 7 results in Mathlib
- Foundations
- Depth 122 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
- pathComponentInstatement and proof · cited by 19
- JoinedIn.target_memproof · cited by 2
Cited by7
Results whose statement or proof uses this declaration.
- IsOpen.pathComponentInproof · cited by 5
- locallyPathConnectedSpace_iff_isOpen_pathComponentInproof · cited by 3
- LocallyPathConnectedSpace.coinducedproof · cited by 2
- locallyPathConnectedSpace_iff_pathComponentIn_mem_nhdsproof · cited by 1
- ChartedSpace.locallyPathConnectedSpaceproof · cited by 1
- Pi.locallyPathConnectedSpace_of_finite_not_pathConnectedSpaceproof · cited by 1
- isOpen_isPathConnected_basisproof · cited by 0