Theorems · Theorem · general topology
IsOpen.isConnected_iff_isPathConnected
∀ {X : Type u_1} [inst : TopologicalSpace X] [LocallyPathConnectedSpace X] {U : Set X},
IsOpen U → (IsConnected U ↔ IsPathConnected U)- Cited by
- 0 results in Mathlib
- Foundations
- Depth 134 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites12
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.Elemproof · cited by 7,166
- IsOpenstatement and proof · cited by 2,400
- IsConnectedstatement · cited by 116
- IsPathConnectedstatement and proof · cited by 65
- LocallyPathConnectedSpacestatement and proof · cited by 53
- ConnectedSpaceproof · cited by 37
- isConnected_iff_connectedSpaceproof · cited by 7
- isPathConnected_iff_pathConnectedSpaceproof · cited by 4
- IsOpen.locallyPathConnectedSpaceproof · cited by 3
- pathConnectedSpace_iff_connectedSpaceproof · cited by 1
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