given in the next sections. In fact, several $UV$-properties can be expressed in these terms by saying that a subset $S\subset Y$ is $UV^*$ if every neighbourhood $U$ of $S$ contains a neighbourhood $V$ of $S$ such that $V{\lhook\joinrel\xrightarrow{*}} U$.

In what follows, all covers are assumed to be of *nonempty* sets. Also, in all statements in this section, $(Y,\rho)$ is a fixed metric space and ${\lhook\joinrel\xrightarrow{*}}$ is a monotone embedding relation on the subsets of $Y$. In this general setting, we shall say that a subset $S\subset Y$ is *uniformly $UV^*$* if for every $\varepsilon>0$ there exists $\delta=\delta(\varepsilon)>0$ such that $\mathbf{O}_\delta(S){\lhook\joinrel\xrightarrow{*}} \mathbf{O}_\varepsilon(S)$. Clearly, we may always pick $\delta$ such that $\delta(\varepsilon)\leq \varepsilon$, which is in good accord with the requirement that $\mathbf{O}_\delta(S)\subset \mathbf{O}_\varepsilon(S)$.

The main goal of this section is the following approximative representation of $\rho$-continuous mappings with uniformly $UV^*$-values.

\[theorem-Near-sel-v1:1\] Let $X$ be a paracompact space and $\varphi:X\operatorname{\leadsto}Y$ be a $\rho$-continuous mapping such that each $\varphi(x)$, $x\in X$, is uniformly $UV^*$. Then for every continuous function $\varepsilon:X\to (0,+\infty)$ there exists a sequence $\mathscr{U}_n$, $n<\omega$, of open covers of $X$ and mappings $\Phi_n,\Psi_n:\mathscr{U}_n\operatorname{\leadsto}Y$ such that for every $U\in \mathscr{U}_n$,

1.  \[item:Near-sel-v3:1\] $\Phi_n(U){\lhook\joinrel\xrightarrow{*}}\Psi_n(U)\subset
        \mathbf{O}_{\varepsilon(p)}(\varphi(p))$, whenever $p\in U$

2.  \[item:Near-sel-v3:2\] $\Psi_k(V)\subset 