For compact subspaces $C$ and $K$ of $X$ and $Y$, prove that for every open set $U$ of $X \times Y$, there exist open sets $V$ and $W$ with...

Question

Let $C$ be a compact subspace of $X$ and let $K$ be a compact subspace of $Y$ . Let $U$ be an open set in $X \times Y$ containing $C \times K$. Show that there exist open subspaces $V$ of $X$ and $W$ of $Y$ with $$C \times K \subset V \times W \subset U.$$

Any open set $U$ in $X \times Y$ is of the form $V \times W$ with $V$ open in $X$ and $W$ open in $Y$. By definition, $C \times K \subset U$ implies that $C \subset V$ and $K \subset W$. Since I didn't use the compactness of $C$ and $K$, is there something I'm missing?

Answer 1

You are incorrect that every open set $U$ in $X\times Y$ is of the form $V\times W$ for some $X$-open $V$ and $Y$-open $W.$ Consider for example the open unit disk in the Euclidean plane.

What you can say, however, is that every such open set $U$ is a union of sets of that form--or put another way, sets of that form comprise a basis for the product topology on $X\times Y.$

For an outline of a proof, see here.