# Prove that if $X \times Y$ is compact, so are $X$ and $Y$.

For the opposite direction, given that X and Y are compact we start with taking $O$ to be an open cover for $X \times Y$. Then for each $x \in X$, ${x} \times Y$ is compact. Later to conclude that the union of open sets $O_x$ is open and for each $x \in X$ there exists an open $W_x$ $\subset X$ such that $W_x \times Y$ is covered by $O_x$. Along those lines... but how do I go backwards?

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The answer below gives a more elegant solution, but starting with an open cover of $X$, what is the only open cover of $X\times Y$ that comes to mind that can be created from it? –  Carsten Schultz Nov 4 '13 at 11:13