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 $U\times V\subset W$.
I feel that I am misunderstanding the usage of intersections here. In the first case, I thought we define $U = U_{b_1} \cap ... \cap U_{b_n}$ because ${a}$ is a point that is part of every $U_{b_i}$ but in the following case when A is an arbitrary compact set we define $V$ using intersections too. Can anyone explain why?
In the first case, you're right. $B$ is contained in every $V_a$ so it's contained in their intersection, $V$. We require a union in the $U_a$ because we need $U$ to cover all of $A$, not just some particular $a$.