characterizations of a ω-cover in a Tychonoff space by co-zero sets

Question

My questions is: 1. How to characterize an $\omega$-cover in a Tychonoff space by co-zero sets. (An open cover $\mathcal{U}$ is a $\omega$-cover of $X$ if $X\not\in \mathcal{U}$ and for any finite subset $F$ of $X$ there is a $U\in \mathcal{U}$ such that $F\subseteq U$.)