= Solution
A <homeomorphism> is a <bijection> for which both the <function> and its inverse are <continuous maps>.
Let $f:X\to Y$ satisfy the given assumptions. For a <closed subset> $F$ of $X$, the <closed subset of a compact space> theorem makes $F$ <compact>. The <continuous image of a compact space> theorem makes $f(F)$ <compact>. Because $Y$ is <Hausdorff>, its <compact subset> $f(F)$ is <closed>. Thus $f$ is a <closed map>.
For $g=f^{-1}:Y\to X$, the <inverse image> of a <closed set> $F\subseteq X$ is
$$
g^{-1}(F)=f(F),
$$
which is <closed> in $Y$. The <closed-set criterion for continuity> therefore makes $g$ <continuous>. Hence \b[$f$ is a <homeomorphism>], proving the <compact-to-Hausdorff continuous bijection theorem>.
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