A <homeomorphism> $f:M\to N$ is a <bijection> for which both $f$ and $f^{-1}$ are <continuous maps>, using the respective <metric topologies>.
$$
\boxed{f\text{ homeomorphism}\iff f\text{ bijective and }f,f^{-1}\text{ continuous}.}
$$
<Continuity> of a <bijection> alone is insufficient: <continuity> of the inverse is a separate requirement.

Solution

Past exam of the metric spaces and complex analysis course of the University of Oxford / 2014 / 1 / a / i / Solution

 Solution (source code)