Compactness of graphs lemma

The graph of a continuous map from a compact space is compact in the product.

Compactness of graphs lemma

Compactness of graphs lemma: Let $f:X\to Y$ be a continuous map between topological spaces, and let

\Gamma_f=\{(x,f(x)):\ x\in X\}\subseteq X\times Y

be its graph, where $X\times Y$ carries the product topology . If $X$ is compact , then $\Gamma_f$ is a compact subset of $X\times Y$ . If, in addition, $Y$ is Hausdorff , then $\Gamma_f$ is a closed set in $X\times Y$ .

This observation is often combined with the fact that compact subsets of Hausdorff spaces are closed (see compact subsets of Hausdorff spaces are closed ) in arguments about homeomorphisms .