for any sequence $\{\mathscr{U} _n:n<\omega\}$ of open covers of $X$ there exists a finite sequence $\{\mathscr{V} _n:n\leq k\}$ of discrete open families in $X$ such that each $\mathscr{V} _n$ refines $\mathscr{U} _n$ and $\bigcup_{n\leq k}\mathscr{V} _n$ is a cover of $X$.

Regarding the proper place of finite $C$-spaces, it was shown by Valov in [@valov:00 Proposition 2.2] that a space $X$ is a finite $C$-space if and only if its Čech-Stone compactification $\beta X$ is a $C$-space. Accordingly, each compact $C$-space is a finite $C$-space. On the other hand, there are $C$-spaces whose Čech-Stone compactification is not a $C$-space, see e.g.[@MR2080284 Remark 3.7].

In this section, we extend Theorem \[theorem-Selections-IDS-v4:1\] to all paracompact finite $C$-spaces. It is based on the relation “$S{\lhook\joinrel\xrightarrow{\omega}} T$” defined in Section \[sec:orient-simpl-compl\], which is another example of a monotone embedding relation. In terms of this relation, we shall say that a subset $S$ of a metric space $(Y,\rho)$ is *uniformly $UV^\omega$* if for each $\varepsilon>0$ there exists $\delta\in (0,\varepsilon]$ such that $\mathbf{O}_\delta(S){\lhook\joinrel\xrightarrow{\omega}} \mathbf{O}_\varepsilon(S)$.

\[theorem-Finite-C-v5:1\] Let $X$ be a paracompact finite $C$-space, $(Y,\rho)$ be a metric space and $\varphi:X\operatorname{\leadsto}Y$ be a $\rho$-continuous mapping such that each $\varphi(x)$, $x\in X$, is uniformly $UV^\omega$. Then $\varphi$ has a 