Theorems · Theorem · general topology
mem_pathComponentIn_self
∀ {X : Type u_1} [inst : TopologicalSpace X] {x : X} {F : Set X}, x ∈ F → x ∈ pathComponentIn F x- Defined in
- Mathlib.Topology.Connected.PathConnected
- Cited by
- 7 results in Mathlib
- Foundations
- Depth 121 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 · cited by 19
- JoinedIn.reflproof · cited by 3
Cited by7
Results whose statement or proof uses this declaration.
- isPathConnected_pathComponentInproof · cited by 6
- locallyPathConnectedSpace_iff_isOpen_pathComponentInproof · cited by 3
- LocallyPathConnectedSpace.coinducedproof · cited by 2
- pathComponentIn_mem_nhdsproof · cited by 2
- locallyPathConnectedSpace_iff_pathComponentIn_mem_nhdsproof · cited by 1
- pathComponentIn_nonempty_iffproof · cited by 0
- isOpen_isPathConnected_basisproof · cited by 0