Theorems · Theorem · general topology
IsOpen.pathComponentIn
∀ {X : Type u_1} [inst : TopologicalSpace X] {F : Set X} [LocallyPathConnectedSpace X],
IsOpen F → ∀ (x : X), IsOpen (pathComponentIn F x)- Cited by
- 5 results in Mathlib
- Foundations
- Depth 129 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites16
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
- nhdsproof · cited by 5,554
- IsOpenstatement and proof · cited by 2,400
- IsOpen.mem_nhdsproof · cited by 470
- Filter.mem_of_supersetproof · cited by 308
- Filter.HasBasis.mem_iffproof · cited by 193
- mem_of_mem_nhdsproof · cited by 126
- IsPathConnectedproof · cited by 65
- LocallyPathConnectedSpacestatement and proof · cited by 53
- isOpen_iff_mem_nhdsproof · cited by 48
- pathComponentInstatement and proof · cited by 19
Cited by5
Results whose statement or proof uses this declaration.
- locallyPathConnectedSpace_iff_isOpen_pathComponentInproof · cited by 3
- LocallyPathConnectedSpace.coinducedproof · cited by 2
- IsOpen.pathComponentproof · cited by 2
- pathComponentIn_mem_nhdsproof · cited by 2
- isOpen_isPathConnected_basisproof · cited by 0