A homeomorphism $f$ of topological spaces is a continuous, bijective map such that $f^{-1}$ is also continuous. We also say that two spaces are homeomorphic if such a map exists.

If two topological spaces are homeomorphic, they are topologically equivalent -- using the techniques of topology, there is no way of distinguishing one space from the other.

An autohomeomorphism (also known as a self-homeomorphism) is a homeomorphism from a topological space to itself.