if $U\cap |\sigma|$ is open in $|\sigma|$, for every $\sigma\in \Sigma$. The following special case of Theorem \[theorem-Selections-IDS:1\] was obtained by Haver in [@haver:1974 Proposition 3].

\[theorem-Selections-IDS-v4:1\] Let $X$ be a compact metric $C$-space, $(Y,\rho)$ be a metric space and $\varphi:X\operatorname{\leadsto}Y$ be a $\rho$-continuous mapping such that for every $x\in X$ and $\varepsilon>0$, there exists $\delta>0$ with the property that for every finite simplicial complex $\Sigma$, every continuous map $g:|\Sigma|\to \mathbf{O}_\delta(\varphi(x))$ is null-homotopic in $\mathbf{O}_\varepsilon(\varphi(x))$. Then $\varphi$ has a continuous $\varepsilon$-selection, for every $\varepsilon>0$.

In the present paper, we show that Theorem \[theorem-Selections-IDS:1\] is valid for all paracompact $C$-spaces, while Theorem \[theorem-Selections-IDS-v4:1\] — for all paracompact finite $C$-spaces. In fact, both these generalisations are obtained with an almost identical proof. Namely, in the next section, we consider an abstract embedding relation $S{\lhook\joinrel\xrightarrow{*}}T$ of subsets $S\subset T\subset Y$ of a space $Y$. The only requirement we place on this relation is to be *monotone* in the sense that $A\subset S{\lhook\joinrel\xrightarrow{*}} T\subset B$ implies $A{\lhook\joinrel\xrightarrow{*}} B$. In this general setting, for a metric space $(Y,\rho)$, we consider *uniformly $UV^{*}$* subsets defined by “$\mathbf{O}_\delta(S){\lhook\joinrel\xrightarrow{*}} \mathbf{O}_\varepsilon(S)$”, i.e.precisely as the properties of the values of $\varphi$ in Theorems \[theorem-Selections-IDS:1\] and \[theorem-Selections-IDS-v4:1\], but now using the 