Prove that if $A, B$ are compact sets in topological spaces $X,Y$ respectively s.t $A\times B$ is contained an open set $N$ of the product topology $X\times Y$ then there exists open sets $U\in X$ and $V\in Y$ s.t $A\times B\subset U\times V\subset N$.
Actually I already proved this problem. But I still want an example to show that if we eliminate one of the compactness of $A,B$, the statement is not true. Could someone give me an example? Thanks in advance.