see [@gutev:2018a Proposition 2.5], $g$ is a continuous map such that $g(x)\in |\Sigma_\mathscr{V}(x)|$, $x\in X$, where $\Sigma_\mathscr{V}(x)=\big\{\sigma\in \mathscr{N}(\mathscr{V}):
  x\in \bigcap\sigma\big\}$. The composite map $f=h\circ g$

&()\
[X]{} & Y

is now a continuous $\varepsilon$-selection for $\varphi$. Indeed, take $x\in X$ and a simplex ${\sigma\in \Sigma_\mathscr{V}(x)}$ with $g(x)\in |\sigma|$. Then $x\in \min \sigma$ and by and , $$f(x)=h(g(x))\in h(|\sigma|)\subset
    \Psi(\min \sigma)\subset
    \mathbf{O}_{\varepsilon(x)}(\varphi(x)).\qedhere$$

Approximate Selections and Finite $C$-spaces {#sec:appr-select-finite}
============================================

Finite $C$-spaces were defined by Borst [@MR2280911] for separable metrizable spaces, subsequently the definition was extended by Valov [@valov:00] for arbitrary spaces. For simplicity, we will consider these spaces in the realm of normal spaces. In this setting, a (normal) space $X$ is called a *finite $C$-space* if for any sequence $\{\mathscr{U} _n:n<\omega\}$ of finite open covers of $X$ there exists a finite sequence $\{\mathscr{V} _n:n\leq k\}$ of pairwise disjoint open families in $X$ such that each $\mathscr{V} _n$ refines $\mathscr{U} _n$ and $\bigcup_{n\leq k}\mathscr{V} _n$ is a cover of $X$. The following characterisation of paracompact finite $C$-spaces was obtained Valov in [@valov:00 Theorem 2.4], it brings the defining property of these spaces closer to that of paracompact $C$-spaces.

\[theorem-Selections-IDS-v2:2\] A paracompact space $X$ is a finite $C$-space if and only if 