I'm preparing for a topology prelim, and there's one question related to compactness that I'm trying to work on. Here it is:
Let $C$ be a compact subspace of $X$, and $K$ be a compact subspace of $Y$. Now let $U$ be an open set in $X \times Y$ that contains $C \times K$. Show that there exist open subspaces $V$ of $X$ and $W$ of $Y$ such that $C \times K \subset V \times W \subset U.$ Well, certainly $C \times K$ is a compact subspace of $X \times Y$, right? Then this would mean that every covering of $C \times K$ by open sets in $X \times Y$ contains a finite subcollection covering $C \times K$. Can't you use $U$ as the open covering, and then somehow get a finite subcollection which might have something to do with the space $V \times W$? What am I missing here? I would appreciate some helpful input, thanks.
My second attempt:
Alright, I might as well let $U = A \times B$, $A,B$ open such that $C \subset A, K \subset B$. Now we could let $\mathcal{C}$ be an open covering of $C$; since $C$ is compact, there is a finite subcollection $\{V_1, V_2, … ,V_n\}$ of $\mathcal{C}$ covering $C$. Next, let $V= A \cap (\bigcup_{i=1}^n V_i)$. Then, $V \subset A$, $V$ open and $C \subset V$. Similarly, we can find $W$ such that $W \subset B$, $W$ open and $K \subset W$ so that $C \times K \subset V \times W \subset A \times B= U.$ I hope this was the right approach to take.