Tychonoff reflection in products and the $\omega$-topology on function spaces

We show that if $X$ is a topological space such that $CR(X)$, the topology on $X$ generated by the cozero sets, is not locally compact, then there is a regular space $Y$ such that $CR(X \times Y) \ne CR(X) \times CR(Y)$. We use the $\omega$-topology on the space of continuous functions $C\left( {X,Y} \right)$ (where $\omega$ is an open cover of $X$) which was defined by Arens and Dugundji in 1950.