I am reading a paper. The link is as follows: https://www.sciencedirect.com/science/article/abs/pii/S0166864109005124.
I am trying to understand the proof of (4) →(1)' of Theorem 54. To proceed, I start with a $\omega$-cover $~\cal U$ of $Y$. As $Y$ is Tychonoff there is a refinement $\cal V$ of $\cal U$ consists of co-zero sets which covers $Y$. There is a countable subcollection of $\cal V$, viz. $\cal W$ covers $Y$. $\{f^{-1}(W):W\in \cal W\}$ is a cover of $X$ by co-zero sets. But this is not necessarily a $\omega$-cover.
Then how to use (4) to reach to (1)'?