The space X is said to be path-connected if for any two points x and y in X there exists a continuous function f from the unit interval [0,1] to X with f(0) = x and f(1) = y. (This function is called a path, or curve, from x to y.)
Every path-connected space is connected. Example of connected spaces that are not path-connected include the extended long line L* and the topologist's sine curve. The latter is a certain subset of the Euclidean plane:
If X and Y are topological spaces, f is a continuous function from X to Y, and X is connected (respectively, path-connected), then the image f(X) is connected (respectively, path-connected). The intermediate value theorem can be considered as a special case of this result.
The maximal nonempty connected subsets of any topological space are called the components of the space. The components form a partition of the space (that is, they are disjoint and their union is the whole space). Every component is a closed subset of the original space. The components in general need not be open: the components of the rational numbers, for instance, are the one-point sets. A space in which all components are one-point sets is called totally disconnected.
A topological space is said to be locally connected if it has a base of connected sets. It can be shown that a space X is locally connected if and only if every component of every open set of X is open. The topologist's sine curve shown above is an example of a connected space that is not locally connected.
Similarly, a topological space is said to be locally path-connected if it has a base of path-connected sets. An open subset of a locally path-connected space is connected if and only if it is path-connected. This generalizes the earlier statement about Rn and Cn, each of which is locally path-connected. More generally, any topological manifold is locally path-connected.