## Path Connectedness

A topological space is path connected if for each pair of points there exists a continuous function from the unit interval to with and Path connectedness is an equivalence relation: since the constant map is continuous. since if is continuous with we can write then and with continuous by continuity of  and since if there exists and with then we can define  is continuous since and so there exists a path in from to and This equivalence relation splits into 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 since but there is no continuous function in with but  is path connected however.

Subsets of the real line are connected if and only if they are path-connected; these subsets are the intervals of Also, open subsets of or are 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. 