necessary condition for a $T_1$ space to be paracompact

Question

If a $T_1$ space is paracompact then for every open cover $\{U_s:s\in S\}$ of $X$ there is a continuous mapping $f:X\rightarrow Y$ onto a metrizable space $Y$ and an open cover $\{W_s:s\in S\}$ of $Y$ such that $f^{-1}(W_s)\subset U_s$ for every $s\in S.$ How to prove it?

Answer 1

I think the following can work: let $\{f_s: X \to [0,1]: s \in S\}$ be a partition of unity subordinated to $\{U_s: s \in S\}$. Then define a map to the hedghehog space of spinyness $|S|$ and pull back stalks.