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:

- { (
*x*,*y*) in**R**^{2}| 0 <*x*and*y*= sin(1/*x*) } union { (0,*y*) in**R**^{2}| -1 ≤*y*≤ 1 }.

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 **R**^{n} and **C**^{n}, each of which is locally path-connected.
More generally, any topological manifold is locally path-connected.