If a paracompact space is a countable union of closed $C$-subspaces, then it is itself a $C$-space [@gutev-valov:00 Theorem 4.1], see also [@addis-gresham:78 Theorem 2.7]. Accordingly, the space $X$ in Theorem \[theorem-Selections-IDS:1\] is a $C$-space. In this section, we show that Theorem \[theorem-Selections-IDS:1\] remains valid for all paracompact $C$-spaces. It is based on the relation “$S{\lhook\joinrel\xrightarrow{\infty}}T$” defined in the previous section, which is clearly a monotone embedding relation. In its terms, a subset $S$ of a metric space $(Y,\rho)$ is *CE* iff it is *uniformly $UV^\infty$* (see Section \[sec:monot-embedd-relat\]), namely if for every $\varepsilon>0$ there exists $\delta\in (0,\varepsilon]$ such that $\mathbf{O}_\delta(S){\lhook\joinrel\xrightarrow{\infty}} \mathbf{O}_\varepsilon(S)$.

\[theorem-Finite-C-v6:1\] Let $X$ be a paracompact $C$-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 $UV^\infty$. Then $\varphi$ has a continuous $\varepsilon$-selection, for every continuous function ${\varepsilon:X\to (0,+\infty)}$.

For a continuous function $\varepsilon:X\to (0,+\infty)$, take open covers $\mathscr{U}_n$, $n<\omega$, of $X$ and mappings $\Phi_n,\Psi_n:\mathscr{U}_n\operatorname{\leadsto}Y$ as those in Theorem \[theorem-Near-sel-v1:1\] (applied with the relation ${\lhook\joinrel\xrightarrow{\infty}}$ and the function $\varepsilon$). Then, since $X$ is a $C$-space, there exists a sequence $\mathscr{V}_{n}$, $n<\omega$, of pairwise disjoint families of nonempty open subsets of $X$ such that each $\mathscr{V}_n$, $n<\omega$, refines $\mathscr{U}_n$, 