Closed graph of a map into a Hausdorff space
= Closed graph of a map into a Hausdorff space
For a <continuous map> $f:X\to Y$ with <Hausdorff space> $Y$, its <graph of a function> is closed in the <product topology>. If $y\ne f(x)$, choose disjoint neighborhoods of those two points in $Y$. Continuity supplies a neighborhood of $x$ whose image lies in the neighborhood of $f(x)$; its product with the other neighborhood misses the graph.