Theorems · Theorem · general topology
isPathConnected_iff_pathConnectedSpace
∀ {X : Type u_1} [inst : TopologicalSpace X] {F : Set X}, IsPathConnected F ↔ PathConnectedSpace ↑F- Defined in
- Mathlib.Topology.Connected.PathConnected
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 131 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.
Cites11
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
- Set.Elemstatement · cited by 7,166
- Set.image_univproof · cited by 322
- IsPathConnectedstatement and proof · cited by 65
- Topology.IsInducing.subtypeValproof · cited by 43
- PathConnectedSpacestatement · cited by 26
- Set.ofPred_mem_eqproof · cited by 16
- Subtype.range_val_subtypeproof · cited by 8
- pathConnectedSpace_iff_univproof · cited by 7
- Topology.IsInducing.isPathConnected_iffproof · cited by 4
Cited by4
Results whose statement or proof uses this declaration.
- IsSimplyConnected.isPathConnectedproof · cited by 1
- Metric.isPathConnected_closedEBallproof · cited by 0
- isSimplyConnected_iff_exists_homotopy_refl_forall_memproof · cited by 0
- IsOpen.isConnected_iff_isPathConnectedproof · cited by 0