be a metric space and $X=\bigcup_{n<\omega} X_n$ be a metrizable space with each $X_n$ being a compact $C$-space. Then each $\rho$-continuous CE-valued mapping $\varphi:X\operatorname{\leadsto}Y$ has a continuous $\varepsilon$-selection, for every continuous function $\varepsilon:X\to (0,1]$.

By a *simplicial complex* we mean a collection $\Sigma$ of nonempty finite subsets of a set $S$ such that $\tau\in \Sigma$, whenever ${\varnothing}\neq \tau\subset \sigma\in \Sigma$. The set $\bigcup\Sigma $ is the *vertex set* of $\Sigma$, while each element of $\Sigma$ is called a *simplex*. The *$k$-skeleton* $\Sigma^k$ of $\Sigma$ ($k\geq 0$) is the simplicial complex $\Sigma^{k}=\{\sigma\in \Sigma:\operatorname{Card}(\sigma)\leq k+1\}$, where $\operatorname{Card}(\sigma)$ is the cardinality of $\sigma$. In the sequel, for simplicity, we will identify the vertex set of $\Sigma$ with its $0$-skeleton $\Sigma^0$.

The vertex set $\Sigma^0$ of a simplicial complex $\Sigma$ can be embedded as a linearly independent subset of some linear normed space. Then to any simplex $\sigma\in \Sigma$ we may associate the *geometric simplex* $|\sigma|$ which is the convex hull of $\sigma$. The *geometric realisation* of $\Sigma$ is the set $|\Sigma|=\bigcup\{|\sigma|:\sigma\in \Sigma\}$. As a topological space, we will always consider $|\Sigma|$ endowed with the *Whitehead topology* [@MR1576810; @MR0030759]. This is the topology in which a subset $U\subset |\Sigma|$ is open if and only 