= Compact graph of a continuous map
If $X$ is a <compact space> and $f:X\to Y$ is a <continuous map>, the <graph of a function> of $f$ is <compact> in the <product topology> on $X\times Y$. Indeed,
$$
G:X\longrightarrow X\times Y,\qquad G(x)=(x,f(x))
$$
is a <continuous map> onto that graph, so the result follows from the <continuous image of a compact space> theorem. No <Hausdorff> assumption on $Y$ is needed.
The coordinate <projection map> restricted to the graph is a continuous inverse to $G$. Thus the graph is actually <homeomorphic> to $X$. For instance, the graph of $x^2$ over $[0,1]$ has the same intrinsic <topology> as the interval, even though it is curved in the plane.
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