relation ${\lhook\joinrel\xrightarrow{*}}$. Next, we show that any $\rho$-continuous mapping $\varphi:X\operatorname{\leadsto}Y$ with uniformly $UV^*$-values, admits a special sequences $\{\mathscr{U}_n:n<\omega\}$ of open covers of $X$ approximating the values of $\varphi$, see Theorem \[theorem-Near-sel-v1:1\]. This is essentially the only place where $\rho$-continuity of $\varphi$ is used. Section \[sec:orient-simpl-compl\] contains two constructions of continuous maps from the geometric realisation of oriented simplicial complexes, see Theorems \[theorem-Near-sel-v7:1\] and \[theorem-Near-sel-v9:1\]. Finally, in Section \[sec:appr-select-c\], we apply Theorem \[theorem-Near-sel-v7:1\] to nerves of covers to obtain with ease our generalisation of Theorem \[theorem-Selections-IDS:1\] (see Theorem \[theorem-Finite-C-v6:1\]). The same approach, but now using Theorem \[theorem-Near-sel-v9:1\], gives the generalisation of Theorem \[theorem-Selections-IDS-v4:1\] (see Theorem \[theorem-Finite-C-v5:1\]). The last section of the paper contains two applications of these theorems dealing with near-selections for Hausdorff continuous mappings whose values are absolute retracts, see Corollaries \[corollary-Near-sel-vgg:1\] and \[corollary-Near-sel-vgg:2\].

Monotone Embedding Relations {#sec:monot-embedd-relat}
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Let $S{\lhook\joinrel\xrightarrow{*}}T$ be an embedding relation defined on subsets $S,T\subset Y$ of a space $Y$. In this, we will always assume that $S\subset T$, so that the relation ${\lhook\joinrel\xrightarrow{*}}$ places some extra properties on the inclusion $S\subset T$. Such a relation will be called *monotone* if $A\subset S{\lhook\joinrel\xrightarrow{*}} T\subset B$ implies $A{\lhook\joinrel\xrightarrow{*}} B$. Various examples of monotone embedding relations will be 