> [!definition] > > Let $(X, \topo_X)$ and $(Y, \topo_Y)$ be [[Topological Space|topological spaces]]. A mapping $f: X \to Y$ is a **homeomorphism** if it is bijective with $f$ and $f^{-1}$ both [[Continuity|continuous]], in which case it induces a bijection between $\topo_X$ and $\topo_Y$. > > If there is a homeomorphism between $X$ and $Y$, then $X$ and $Y$ are **homeomorphic**. > [!theorem] > > Let $X$ be a [[Compactness|compact]] space and $Y$ be a [[Hausdorff Space|Hausdorff]] space, then any continuous bijection $f: X \to Y$ is a homeomorphism. > > *Proof*. Let $E \subset X$ be [[Closed Set|closed]], then it is also compact. As $f$ is continuous, $f(E)$ is compact as well. With $Y$ being Hausdorff, $f(E)$ is [[Closed Set|closed]]. Therefore $f(E^c)$ is open in $Y$, and $f^{-1}$ is continuous.