Theorems · Theorem · general topology
isConnected_iff_connectedSpace
∀ {α : Type u} [inst : TopologicalSpace α] {s : Set α}, IsConnected s ↔ ConnectedSpace ↑s- Defined in
- Mathlib.Topology.Connected.Basic
- Cited by
- 7 results in Mathlib
- Foundations
- Depth 81 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.
Cites8
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 and proof · cited by 7,166
- IsConnectedstatement · cited by 116
- ConnectedSpacestatement and proof · cited by 37
- nonempty_subtypeproof · cited by 22
- isPreconnected_iff_preconnectedSpaceproof · cited by 4
- Subtype.connectedSpaceproof · cited by 2
Cited by7
Results whose statement or proof uses this declaration.
- IsPathConnected.isConnectedproof · cited by 7
- AffineSubspace.isConnected_setOfPred_sSameSideproof · cited by 3
- AffineSubspace.isConnected_setOfPred_sOppSideproof · cited by 2
- AffineSubspace.isConnected_setOfPred_wOppSideproof · cited by 2
- AffineSubspace.isConnected_setOfPred_wSameSideproof · cited by 2
- IsOpenMap.finite_connectedComponents_of_finite_preimage_singletonproof · cited by 0
- IsOpen.isConnected_iff_isPathConnectedproof · cited by 0