A topological spaceis path connected if for each pair of pointsthere exists a continuous functionfrom the unit intervaltowithand
Path connectedness is an equivalence relation:
since the constant mapis continuous.
since ifis continuous withwe can writethenandwithcontinuous by continuity of
andsince if there existsandwiththen we can define
is continuous sinceandso there exists a path infromtoand
This equivalence relation splitsinto path connected components.
Every path-connected space is connected. The converse is not always true. The complex plane with infinity included is not path connected sincebut there is no continuous function in withbutis path connected however.
Subsets of the real lineare connected if and only if they are path-connected; these subsets are the intervals ofAlso, open subsets oforare connected if and only if they are path-connected. Connectedness and path-connectedness are the same for finite topological spaces.
An open set may or may not be path connected, but the closure of an open connected set is also path connected, since all closure does is include the limit points. Path connectedness is a topological property preserved by homeomorphisms. In particular, a homeomorphisms preserves the number of path connected components.