finite-dimensional paracompact space, as well as every countable-dimensional metrizable space, is a $C$-space [@addis-gresham:78], but there exists a compact metric $C$-space which is not countable-dimensional [@pol:81].

For spaces (sets) $X$ and $Y$, we write $\varphi:X\operatorname{\leadsto}Y$ to designate that $\varphi$ is a *set-valued* (or *multi-valued*) *mapping* from $X$ to the nonempty subsets of $Y$. For a metric space $(Y,\rho)$ and $\varepsilon>0$, we will use $\mathbf{O}_\varepsilon(y)$ for the open $\varepsilon$-ball centred at a point $y\in Y$; and $\mathbf{O}_\varepsilon(S)=\bigcup_{q\in S}\mathbf{O}_\varepsilon(q)$, whenever $S\subset Y$. In these terms, a set-valued mapping $\varphi:X\operatorname{\leadsto}Y$ is *$\rho$-continuous* (or *Hausdorff continuous*) if for every $\varepsilon >0$, each point $x\in X$ has a neighbourhood $V$ such that $$\label{eq:Finite-C-v2:3}
  \varphi(x)\subset \mathbf{O}_\varepsilon(\varphi(p))\ \text{and}\ 
  \varphi(p) \subset \mathbf{O}_\varepsilon(\varphi(x)),\quad
  \text{for every $p\in V$.}$$ A map $f:X\to Y$ is an *$\varepsilon$-selection* for a mapping $\varphi :X\operatorname{\leadsto}Y$ if $f(x)\in \mathbf{O}_\varepsilon\left(\varphi(x)\right)$, for every $x\in X$. More generally, for a function ${\varepsilon:X\to (0,+\infty)}$, we say that $f:X\to Y$ is an *$\varepsilon$-selection* for $\varphi$ if $f(x)\in \mathbf{O}_{\varepsilon(x)}\left(\varphi(x)\right)$, for every $x\in X$. Finally, following Haver [@MR503224], a subset $S\subset Y$ is called *CE* if for every $\varepsilon>0$ there exists $\delta>0$ such that $\mathbf{O}_\delta(S)$ is contractible in $\mathbf{O}_\varepsilon(S)$. The following theorem was proved by Haver in [@MR503224 Theorem 1].

\[theorem-Selections-IDS:1\] Let $(Y,\rho)$ 