 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,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.