A simple question. Suppose $\{U_{\alpha}\}_{\alpha \in A} $ is a fundamental cover of $X$, that is $V \subset X$ is open if (and only if) $V \cap U_{\alpha} $ is open in $U_{\alpha}$ for all $\alpha \in A$. Take a space $Y$, that you can assume to be compact. Is $\{ U_{\alpha} \times Y\}_{\alpha \in A } $ a fundamental cover of $X \times Y$ ?
I think the answer is "yes". You can also assume that $X,Y$ are metric spaces if you wish to.