Theorem. Let $A\subset X,B\subset Y$ be compact subspaces of topological spaces $X,Y$. Let $A\times B\subset W\subset X\times Y$ with $W\subset X\times Y$ open. Then there exists opens $A\subset U\subset X,B\subset V\subset Y$ such that $U\times V\subset W$.
All proofs I could find are the same: we use the openness of $W$ to get for each point $(a,b)\in A\times B$ an open rectangle $U_{(a,b)}\times V_{(a,b)}\subset W$, and then draw a potato which leads us to apply the compactness of $A,B$ to produce a rectangle $U\times V$.
Question. Is it possible to prove this theorem without mentioning points?
That $W\subset X\times Y$ is open tells us that $A\times B$ admits a basic open cover (i.e by open rectangles) which is contained entirely in $W$. However, without using the points of $A\times B$ to index these open rectangles, I don't see how to proceed.