Mathlib Map

Theorems · Definition · general topology

Joined

{X : Type u_1} → [TopologicalSpace X] → X → X → Prop

The relation "being joined by a path". This is an equivalence relation.

Defined in
Mathlib.Topology.Connected.PathConnected
Cited by
29 results in Mathlib
Foundations
Depth 2 from the axioms · uses no axioms
Assumes
TopologicalSpace
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Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

pathComponent · cited by 30pathComponentJoined.somePath · cited by 12Joined.somePathpathConnectedSpace_iff_univ · cited by 7pathConnectedSpace_iff_un…Joined.trans · cited by 4Joined.transJoined.refl · cited by 3Joined.reflJoined.symm · cited by 3Joined.symmpathSetoid · cited by 2pathSetoidJoined.map · cited by 2Joined.mapJoined.mul · cited by 2Joined.mulJoinedIn.joined · cited by 2JoinedIn.joinedmem_pathComponent_iff · cited by 2mem_pathComponent_iffPathConnectedSpace.casesOn · cited by 1PathConnectedSpace.casesOnPathConnectedSpace.joined · cited by 1PathConnectedSpace.joinedconnectedComponent_eq_iff_joined · cited by 1connectedComponent_eq_iff…selfAdjoint.joined_one_expUnitary · cited by 1selfAdjoint.joined_one_ex…TopologicalSpace · cited by 24529TopologicalSpacePath · cited by 318PathJoinedCITED BYCITES

Cites2

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Cited by34

Results whose statement or proof uses this declaration.