Characterization of ω-cover by co-zero sets.

Question

In a Tychonoff space X if each ω-cover contains a refinement consists of co-zero sets which is again a ω-cover?

Answer 1

Of course:

For each finite set $F$ of $X$ there is some $U(F) \in \mathcal{U}$ with $F \subseteq U(F)$. As $X$ is Tychonoff, there is a cozero set $O(F)$ such that $F \subseteq O(F) \subseteq U(F)$ (e.g. cozero sets are closed under finite unions). Then $\{O(F) : F\in [X]^{<\omega}\}$ is the required refinement and an $\omega$-cover by construction.