---
abstract: 'Haver’s near-selection theorem deals with approximate selections of Hausdorff continuous CE-valued mappings defined on $\sigma$-compact metrizable $C$-spaces. In the present paper, we extend this theorem to all paracompact $C$-spaces. The technique developed to achieve this generalisation is based on oriented simplicial complexes. This approach makes not only a considerable simplification in the proof but is also successful in generalising the special case of compact metric $C$-spaces to all paracompact finite $C$-spaces.'
address: 'Department of Mathematics, Faculty of Science, University of Malta, Msida MSD 2080, Malta'
author:
- Valentin Gutev
title: 'Near-selection theorems for $C$-spaces'
---

Introduction
============

A space $X$ has property $C$, or is a *$C$-space*, if for any sequence $\{\mathscr{U} _n:n<\omega\}$ of open covers of $X$ there exists a sequence $\{\mathscr{V} _n:n<\omega\}$ of pairwise disjoint families of open subsets of $X$ such that each $\mathscr{V} _n$ refines $\mathscr{U} _n$ and $\bigcup_{n<\omega}\mathscr{V} _n$ is a cover of $X$. The $C$-space property was originally defined by W.Haver [@haver:1974] for compact metric spaces, subsequently Addis and Gresham [@addis-gresham:78] reformulated Haver’s definition for arbitrary spaces. It should be remarked that a $C$-space is paracompact if and only if it is countably paracompact and normal, see e.g. [@MR2352366 Proposition 1.3]. Every 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 $(Y,\rho)$ 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 if $U\cap |\sigma|$ is open in $|\sigma|$, for every $\sigma\in \Sigma$. The following special case of Theorem \[theorem-Selections-IDS:1\] was obtained by Haver in [@haver:1974 Proposition 3].

\[theorem-Selections-IDS-v4:1\] Let $X$ be a compact metric $C$-space, $(Y,\rho)$ be a metric space and $\varphi:X\operatorname{\leadsto}Y$ be a $\rho$-continuous mapping such that for every $x\in X$ and $\varepsilon>0$, there exists $\delta>0$ with the property that for every finite simplicial complex $\Sigma$, every continuous map $g:|\Sigma|\to \mathbf{O}_\delta(\varphi(x))$ is null-homotopic in $\mathbf{O}_\varepsilon(\varphi(x))$. Then $\varphi$ has a continuous $\varepsilon$-selection, for every $\varepsilon>0$.

In the present paper, we show that Theorem \[theorem-Selections-IDS:1\] is valid for all paracompact $C$-spaces, while Theorem \[theorem-Selections-IDS-v4:1\] — for all paracompact finite $C$-spaces. In fact, both these generalisations are obtained with an almost identical proof. Namely, in the next section, we consider an abstract embedding relation $S{\lhook\joinrel\xrightarrow{*}}T$ of subsets $S\subset T\subset Y$ of a space $Y$. The only requirement we place on this relation is to be *monotone* in the sense that $A\subset S{\lhook\joinrel\xrightarrow{*}} T\subset B$ implies $A{\lhook\joinrel\xrightarrow{*}} B$. In this general setting, for a metric space $(Y,\rho)$, we consider *uniformly $UV^{*}$* subsets defined by “$\mathbf{O}_\delta(S){\lhook\joinrel\xrightarrow{*}} \mathbf{O}_\varepsilon(S)$”, i.e.precisely as the properties of the values of $\varphi$ in Theorems \[theorem-Selections-IDS:1\] and \[theorem-Selections-IDS-v4:1\], but now using the 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}
============================

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 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.  \[item:Near-sel-v3:2\] $\Psi_k(V)\subset \Phi_n(U)$, whenever $k>n$ and $V\in \mathscr{U}_k$ with $V\cap U\neq {\varnothing}$.

To prepare for the proof, we proceed with several observations.

\[proposition-Near-sel-v1:1\] Let $\varepsilon:X\to (0,+\infty)$ be a continuous function on a space $X$. Then there exists an open cover $\mathscr{V}$ of $X$ and a function $\delta:\mathscr{V}\to (0,+\infty)$ such that $\delta(V)\leq \varepsilon(p)$, for every $p\in V\in \mathscr{V}$.

Since $\varepsilon$ is continuous, each point $x\in X$ is contained in an open set $V_x$ such that $\varepsilon(p)>\frac{\varepsilon(x)}2=\delta_x$, for every $p\in V_x$. Take $\mathscr{V}=\{V_x:x\in X\}$, and for each $V\in \mathscr{V}$ set $\delta(V)=\delta_x$, where $\delta_x$ corresponds to some $x\in X$ with $V=V_{x}$.

Let $\mathscr{F}(Y)$ be the collection of all nonempty closed subsets of $Y$. A natural topology on $\mathscr{F}(Y)$ is the Hausdorff one $\tau_{H(\rho)}$ generated by the Hausdorff distance $H(\rho)$ associated to $\rho$. Recall that the *Hausdorff distance* is defined by $$\begin{aligned}
  \label{eq:Near-sel-v4:1}
  H(\rho)(S,T)&=\inf\big\{\varepsilon>0: S\subset
                \mathbf{O}_\varepsilon(T)\ \text{and}\ T\subset
                \mathbf{O}_\varepsilon(S)\big\}\\
              &=\sup\left\{\rho(S,y)+\rho(y,T): y\in S\cup
                T\right\},\qquad S,T\in \mathscr{F}(Y).\nonumber\end{aligned}$$ The Hausdorff topology $\tau_{H(\rho)}$ is always metrizable even though the distance $H(\rho)$ may assume infinite values. Moreover, given any nonempty set $S\subset Y$, we have that $\rho(y,S)=\rho\left(y,\overline{S}\right)$, for every $y\in Y$. So, we may consider $H(\rho)(S,T)$ defined on all nonempty subsets of $Y$. Then, according to and , a mapping $\varphi:X\operatorname{\leadsto}Y$ is $\rho$-continuous if and only if it is continuous with respect to $H(\rho)$ as a usual map, namely if for every $\varepsilon>0$, each point $x\in X$ is contained in an open set $V$ such that $H(\rho)(\varphi(p),\varphi(x))<\varepsilon$ for every $p\in V$.

\[proposition-Finite-C-v4:1\] Let $\varphi:X\operatorname{\leadsto}Y$ be a $\rho$-continuous mapping on a space $X$, and $x\in X$ be such that $\varphi(x)$ is uniformly $UV^*$. Then for every $\varepsilon>0$ there exists $\delta_x\in (0,\varepsilon)$ and an open set $U_x\subset X$ containing $x$ such that for every $p,q\in U_x$, $$H(\rho)(\varphi(p),\varphi(q))<\frac{\delta_x}2\quad
    \text{and}\quad \mathbf{O}_{\delta_x}(\varphi(p)){\lhook\joinrel\xrightarrow{*}}
    \mathbf{O}_\varepsilon(\varphi(p)).$$

Take $\delta_x\in \left(0,\frac\varepsilon4\right)$ with $\mathbf{O}_{2\delta_x}(\varphi(x)) {\lhook\joinrel\xrightarrow{*}}
  \mathbf{O}_{\frac\varepsilon2}(\varphi(x))$, and set $$U_x=\left\{p\in X:
      H(\rho)(\varphi(p),\varphi(x))<\frac{\delta_x}4\right\}.$$ Then $U_x$ is an open set such that $H(\rho)(\varphi(p),\varphi(q))<\frac{\delta_x}2$, for every $p,q\in U_x$. Moreover, $p\in U_x$ implies $\mathbf{O}_{\delta_x}(\varphi(p)){\lhook\joinrel\xrightarrow{*}}
  \mathbf{O}_\varepsilon(\varphi(p))$ because ${\lhook\joinrel\xrightarrow{*}}$ is monotone and $$\mathbf{O}_{\delta_x}(\varphi(p))\subset
    \mathbf{O}_{2\delta_x}(\varphi(x)) {\lhook\joinrel\xrightarrow{*}}
    \mathbf{O}_{\frac\varepsilon2}(\varphi(x)) \subset
    \mathbf{O}_\varepsilon(\varphi(p)).\qedhere$$

\[proposition-Finite-C-v4:3\] 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 $UV^*$. Suppose that $\mathscr{U}_1$ is a locally finite open cover of $X$ and $\delta_1:\mathscr{U}_1\to (0,+\infty)$. Then there exists an open locally finite refinement $\mathscr{U}_2$ of $\mathscr{U}_1$ and functions $\delta_2,\varepsilon_2:\mathscr{U}_2\to (0,+\infty)$ with $\delta_2\leq \varepsilon_2$, such that

1.  $\mathbf{O}_{\delta_2(U)}(\varphi(p)){\lhook\joinrel\xrightarrow{*}}
        \mathbf{O}_{\varepsilon_2(U)}(\varphi(p))$, for every $p\in U\in \mathscr{U}_2$,

2.  $H(\rho)(\varphi(p),\varphi(q))<\frac{\delta_2(U)}2$, for every $p,q\in U\in \mathscr{U}_2$,

3.  $\varepsilon_2(U_2)\leq \frac{\delta_1(U_1)}3$, whenever $U_i\in \mathscr{U}_i$, $i=1,2$, with $U_1\cap U_2\neq {\varnothing}$.

Since $\mathscr{U}_1$ is locally finite, the paracompact space $X$ has a locally finite open cover $\mathscr{V}$ such that each family $\mathscr{U}_1^V=\left\{U\in \mathscr{U}_1: U\cap \overline{V}\neq
    {\varnothing}\right\}$, $V\in \mathscr{V}$, is finite. For convenience, set $$\label{eq:Near-sel-vgg:1}
    \varepsilon(V)= \min_{U\in
      \mathscr{U}_1^V}\frac{\delta_1(U)}3>0,\quad \text{whenever $V\in
      \mathscr{V}$.}$$ Then by Proposition \[proposition-Finite-C-v4:1\], for each $V\in \mathscr{V}$ there exists an open and locally finite in $\overline{V}$ cover $\mathscr{W}_V$ of the paracompact space $\overline{V}$ and a map $\delta_V:\mathscr{W}_V\to (0,\varepsilon(V))$ such that $\mathscr{W}_V$ refines $\mathscr{U}_1^{V}$ and for every $p,q\in W\in \mathscr{W}_V$, $$\label{eq:Near-sel-vgg:3}
    H(\rho)(\varphi(p),\varphi(q))<\frac{\delta_V(W)}2\quad
    \text{and}\quad \mathbf{O}_{\delta_V(W)}(\varphi(p)){\lhook\joinrel\xrightarrow{*}}
    \mathbf{O}_{\varepsilon(V)}(\varphi(p)).$$

Finally, we can take $\mathscr{U}_2=\left\{W\cap V: W\in \mathscr{W}_V\ \text{and}\ V\in
    \mathscr{V}\right\}$, which is a locally finite open cover of $X$ because so is $\mathscr{V}$ and each $\mathscr{W}_V$, $V\in \mathscr{V}$, is locally finite in $\overline{V}$. To define the required maps $\delta_2,\varepsilon_2:\mathscr{U}_2\to (0,+\infty)$, for each $U\in \mathscr{U}_2$ fix elements $\lambda(U)\in \mathscr{V}$ and $\mu(U)\in \mathscr{W}_{\lambda(U)}$ with $U=\mu(U)\cap \lambda(U)$, and set $$\label{eq:Near-sel-vgg:4}
    \delta_2(U)=\delta_{\lambda(U)}(\mu(U))\quad \text{and}\quad
    \varepsilon_2(U)= \varepsilon(\lambda(U)),\ \text{
      $U\in \mathscr{U}_2$.}$$

If $U_1\cap U_2\neq {\varnothing}$ for some $U_i\in \mathscr{U}_i$, $i=1,2$, then $U_1\cap \lambda(U_2)\neq {\varnothing}$ and, therefore, $U_1\in \mathscr{U}_1^{\lambda(U_2)}$. According to , we get that $\varepsilon_2(U_2)=\varepsilon(\lambda(U_2)) \leq
  \frac{\delta_1(U_1)}3$. The rest of the properties are evident from and . The proof is complete.

We are now ready for the proof of Theorem \[theorem-Near-sel-v1:1\].

Let $\varphi$ and $\varepsilon$ be as in this theorem. Since $X$ is a paracompact space, by Proposition \[proposition-Near-sel-v1:1\], it has an open locally finite cover $\mathscr{V}$ and a function $\delta:\mathscr{V}\to (0,+\infty)$ such that $$\label{eq:Near-sel-v1:1}
    \delta(V)\leq \varepsilon(p),\quad \text{for every
      $p\in V\in \mathscr{V}$.}$$ Hence, by Proposition \[proposition-Finite-C-v4:3\] (applied with $\mathscr{U}_1=\mathscr{V}$ and $\delta_1=\delta$), $X$ has an open locally finite cover $\mathscr{U}_0$ which refines $\mathscr{V}$, and maps $\delta_0,\varepsilon_0: \mathscr{U}_0\to (0,+\infty)$ with $\delta_0\leq \varepsilon_0$, such that for every $p,q\in U\in \mathscr{U}_0$, $$\label{eq:Near-sel-v1:2}
    \begin{cases}
      \mathbf{O}_{\delta_0(U)}(\varphi(p)){\lhook\joinrel\xrightarrow{*}}
      \mathbf{O}_{\varepsilon_0(U)}(\varphi(p)),\\[3pt]
      H(\rho)(\varphi(p),\varphi(q))<\frac{\delta_0(U)}2,\\
      \varepsilon_0(U)\leq \frac{\varepsilon(p)}2.\\
    \end{cases}$$ To see the last property in , take $p\in U\in \mathscr{U}_0$ and $V\in \mathscr{V}$ with $U\subset
  V$. Then by and Proposition \[proposition-Finite-C-v4:3\], $\varepsilon_0(U)\leq \frac{\delta(V)}3\leq
  \frac{\varepsilon(p)}3\leq \frac{\varepsilon(p)}2$.

Using Proposition \[proposition-Finite-C-v4:3\], the construction can be carried on by induction. Thus, there exists a sequence $\mathscr{U}_n$, $n<\omega$, of locally finite open covers of $X$ and functions $\delta_n,\varepsilon_n:\mathscr{U}_n\to (0,+\infty)$ with $\delta_n\leq \varepsilon_n$, such that each $\mathscr{U}_{n+1}$ refines $\mathscr{U}_n$ and the following properties hold for each $p,q\in U\in \mathscr{U}_{n}$: $$\label{eq:Near-sel-v2:1}
    \begin{cases}
      \mathbf{O}_{\delta_n(U)}(\varphi(p)){\lhook\joinrel\xrightarrow{*}}
      \mathbf{O}_{\varepsilon_n(U)}(\varphi(p)),\\[3pt]
      H(\rho)(\varphi(p),\varphi(q))<\frac{\delta_n(U)}2,\\
      \varepsilon_{n+1}(V)\leq \frac{\delta_n(U)}3,\ \text{whenever
        $V\in \mathscr{U}_{n+1}$ with $V\cap U\neq {\varnothing}$.}
    \end{cases}$$

To finish the proof, for each $n<\omega$ take a map $\pi_n:\mathscr{U}_n\to X$ with $\pi_n(U)\in U$, $U\in \mathscr{U}_n$. Next, define $\Phi_n,\Psi_n:\mathscr{U}_n\operatorname{\leadsto}Y$ by $$\label{eq:Near-sel-v5:1}
    \Phi_n(U)=\mathbf{O}_{\delta_n(U)}(\varphi(\pi_n(U)))\quad
    \text{and}\quad
    \Psi_n(U)=\mathbf{O}_{\varepsilon_n(U)}(\varphi(\pi_n(U))),\ U\in
    \mathscr{U}_n.$$

To see \[item:Near-sel-v3:2\], take $k>n$, $U\in \mathscr{U}_n$ and $V_k\in \mathscr{U}_{k}$ with $V_k\cap U\neq {\varnothing}$. Since each $\mathscr{U}_{i+1}$ refines $\mathscr{U}_i$, $i<\omega$, there are elements $V_i\in \mathscr{U}_i$, $n+1\leq i< k$, such that $V_{i+1}\subset V_{i}$ for every $n+1\leq i< k$. Take a point $q\in V_k\cap U\subset V_{n+1}\cap U$. Since $\pi_k(V_k),q\in V_{n+1}$ and $q,\pi_n(U)\in U$, it follows from that $$\begin{aligned}
    H(\rho)(\varphi(\pi_{k}(V_k)),
    &\varphi(\pi_n(U)))\leq\\
    &\leq H(\rho)(\varphi(\pi_{k}(V_k)),\varphi(q))+
      H(\rho)(\varphi(q),\varphi(\pi_n(U)))\\
    &<
      \frac{\delta_{n+1}(V_{n+1})}2+\frac{\delta_n(U)}2
      \leq
      \frac12 \frac{\delta_{n}(U)}3+ \frac{\delta_{n}(U)}2=
      \frac{2\delta_{n}(U)}3.  
  \end{aligned}$$ For the same reason, $\varepsilon_k(V_k)\leq \dots \leq \varepsilon_{n+1}(V_{n+1})\leq
  \frac{\delta_n(U)}3$. Accordingly, $$\begin{aligned}
    \Psi_{k}(V_k)=\mathbf{O}_{\varepsilon_{k}(V_k)}(\varphi(\pi_{k}(V_k)))
    &\subset
      \mathbf{O}_{\frac{\delta_n(U)}3}(\varphi(\pi_{k}(V_{k})))\\
    &\subset
      \mathbf{O}_{\frac{\delta_n(U)}3+
      \frac{2\delta_n(U)}3}(\varphi(\pi_n(U)))=\Phi_n(U).
  \end{aligned}$$

Finally, we show that \[item:Near-sel-v3:1\] holds as well. The “${\lhook\joinrel\xrightarrow{*}}$”-embedding property in \[item:Near-sel-v3:1\] is evident from and . As for the inclusion $\Psi_n(U)\subset \mathbf{O}_{\varepsilon(p)}(\varphi(p))$, where ${p\in U\in \mathscr{U}_n}$, it follows from the special case of $n=0$. Indeed, for $n>0$, take $U_0\in \mathscr{U}_0$ with $U\subset U_0$, which is possible because $\mathscr{U}_n$ refines $\mathscr{U}_0$. Then according to \[item:Near-sel-v3:2\], $\Psi_n(U)\subset \Phi_0(U_0)$ because $U\cap U_0\neq
  {\varnothing}$. Thus, by and , we also have that $$\begin{aligned}
    \Psi_n(U)\subset
    \Phi_0(U_0)&=
                 \mathbf{O}_{\delta_0(U_0)}(\varphi(\pi_0(U_0)))\\
               &\subset \mathbf{O}_{2\delta_0(U_0)}(\varphi(p))\subset
                 \mathbf{O}_{2\varepsilon_0(U_0)}(\varphi(p))\subset 
                 \mathbf{O}_{\varepsilon(p)}(\varphi(p)).\qedhere 
  \end{aligned}$$

Oriented Simplicial Complexes {#sec:orient-simpl-compl}
=============================

As stated in the Introduction, we identify the vertices of a simplicial complex $\Sigma$ with its $0$-skeleton $\Sigma^0$. A simplicial complex $\Sigma$ is called *oriented* if there exists a partial order $\leq$ on its vertex set $\Sigma^0$ such that each simplex of $\Sigma$ is linearly ordered with respect to $\leq$.

The present section deals with two constructions of continuous maps on oriented simplicial complexes. In the first construction, for a space $Y$ and subsets ${S\subset T\subset Y}$, we write $S{\lhook\joinrel\xrightarrow{\infty}} T$ if $S$ is contractible in $T$. Moreover, for mappings $\Phi,\Psi:Z\operatorname{\leadsto}Y$, we will use $\Phi{\lhook\joinrel\xrightarrow{\infty}}\Psi$ to express that $\Phi(z){\lhook\joinrel\xrightarrow{\infty}} \Psi(z)$, for every $z\in Z$.

\[theorem-Near-sel-v7:1\] Let $\Sigma$ be an oriented simplicial complex with respect to some partial order $\leq$ on $\Sigma^0$, and $\Phi,\Psi:\Sigma^0\operatorname{\leadsto}Y$ be mappings into a space $Y$ such that $\Phi{\lhook\joinrel\xrightarrow{\infty}} \Psi$ and $\Psi(v)\subset \Phi(u)$, whenever $u,v\in \Sigma^0$ with $u<v$. Then there exists a continuous map $h:|\Sigma|\to Y$ such that $$\label{eq:Near-sel-v7:1}
    h(|\sigma|)\subset \Psi(\min \sigma),\quad\text{for every
      $\sigma\in \Sigma$.}$$

The proof of Theorem \[theorem-Near-sel-v7:1\] is based on the following considerations. The *cone* $Z*v$ over a space $Z$ with a vertex $v$ is the quotient space of $Z\times[0,1]$ obtained by identifying all points of $Z\times\{1\}$ into a single point $v$. The following representation of $Z*v$ will play an important role in our proof. $$\label{eq:Near-sel-v7:2}
  Z*v=\big\{t v + (1-t) z: z\in Z\ \text{and}\ t\in [0,1]\big\}.$$ This representation is unique for each point $q\in Z*v$ with $q\neq v$, and is in good accord with the fact that $|\sigma|=|\tau|*v$, whenever $\sigma$ is an abstract simplex, $v\in \sigma$ and $\tau=\sigma\setminus\{v\}\neq {\varnothing}$ is the face opposite to the vertex $v$. Finally, for a space $T$ and $S\subset T$, let us recall that a homotopy $H:S\times [0,1]\to T$ is a contraction of $S$ into a point $p\in T$ if $$\label{eq:Near-sel-v7:4}
  H(y,0)=y\ \ \text{and}\ \ 
  H(y,1)=p,\quad \text{for every $y\in S$.}$$

\[proposition-Near-sel-v7:2\] Let $H:S\times [0,1]\to T$ be a contraction of a subset $S\subset T$ into a point $p\in T$ of a space $T$. Then for a space $Z$, each continuous map $g:Z\to S$ can be extended to a continuous map $h:Z*v\to T$ such that $$\label{eq:Near-sel-v7:3}
    h\big(tv+(1-t)z\big)=H(g(z),t),\quad \text{for every $z\in Z$ and
      $t\in [0,1]$.}$$

Define $h:Z*v\to T$ as in , which is possible because $H(g(z),1)=p$ for every $z\in Z$, see and . This $h$ is as required because $H(g(z),0)=g(z)$, for every $z\in Z$.

We apply the above construction to show the special case of Theorem \[theorem-Near-sel-v7:1\] in the setting of an oriented simplex $\sigma$ by a linear order $\leq$ on its vertices. For convenience, such a simplex will be denoted by $(\sigma,\leq)$.

\[proposition-Near-sel-v8:1\] Let $(\sigma,\leq)$ be an oriented simplex and $\Phi,\Psi:\sigma\operatorname{\leadsto}Y$ be mappings into a space $Y$ such that $\Phi{\lhook\joinrel\xrightarrow{\infty}}\Psi$ and $\Psi(v)\subset \Phi(u)$, for every $u,v\in \sigma$ with $u<v$. For every $v\in \sigma$, fix a contraction $H_v:\Phi(v)\times[0,1]\to \Psi(v)$ into some point $p_v\in \Psi(v)$. Then there exists a unique continuous map $h:|\sigma|\to Y$ such that

1.  \[item:Near-sel-v8:1\] $h(v)=p_v$, $v\in \sigma$,

2.  \[item:Near-sel-v8:2\] $h\big(tv+(1-t)z\big)=H_v(h(z),t)$, $z\in |\tau|$ and $t\in [0,1]$, whenever $\tau\subset \sigma$ is a face and $v\in \sigma$ with $v< u$ for every $u\in \tau$.

Suppose that $\sigma=\{v_0,\dots, v_{n+1}\}$, where $v_0<v_1<\dots <v_{n+1}$, and consider the maximal chain $\tau_{n+1}\subset \tau_n\subset\dots \subset \tau_0=\sigma$ of faces of $\sigma$ defined by $\tau_{n+1}=\{v_{n+1}\}$ and $\tau_k=\tau_{k+1}\cup\{v_k\}$, $k\leq n$. We will construct the required map $h:|\sigma|\to Y$ inductively on the faces $|\tau_k|$, $k\leq n+1$, of this chain. To this end, set $S_k=\Phi(v_k)$, $T_k=\Psi(v_k)$, $H_k=H_{v_k}$ and $p_k=p_{v_k}$, $k\leq n+1$. Next, define $h_{n+1}:|\tau_{n+1}|\to T_{n+1}$ by $h_{n+1}(v_{n+1})=p_{n+1}$. Since $|\tau_n|=|\tau_{n+1}|*v_n$ and $T_{n+1}=\Psi(v_{n+1})\subset \Phi(v_n)=S_n$, by Proposition \[proposition-Near-sel-v7:2\], $h_{n+1}$ has a continuous extension $h_n:|\tau_n|\to T_n$ such that $$\begin{aligned}
    h_n\big(tv_n+(1-t)z\big)&=H_n(h_{n+1}(z),t)\\
                            &=H_n(h_n(z),t),\quad\text{$z\in
                              |\tau_{n+1}|$ and $t\in [0,1]$.} 
  \end{aligned}$$ It is also evident that $h_n(v_n)=H_n(p_{n+1},1)=p_n$. The construction can be carried on by a finite induction to get a continuous map $h=h_0:|\tau_0|=|\sigma|\to T_0$ satisfying \[item:Near-sel-v8:1\] and such that for every $0\leq k\leq n$, $$\label{eq:Near-sel-v8:1}
    h\big(tv_k+(1-t)z\big)=H_{k}(h(z),t),\quad \text{$z\in
      |\tau_{k+1}|$ and $t\in [0,1]$.}$$ This $h$ is as required. Indeed, take any continuous map $g:|\sigma|\to Y$ satisfying \[item:Near-sel-v8:1\] and . Then $g(v_{n+1})=h(v_{n+1})$ and by , we get that ${g\operatorname{\upharpoonright}|\tau_n|=h\operatorname{\upharpoonright}|\tau_n|}$. Inductively, this implies that $g=g\operatorname{\upharpoonright}|\tau_0|=h\operatorname{\upharpoonright}|\tau_0|=h$.

To see finally that \[item:Near-sel-v8:2\] is equivalent to , take a face $\tau\subset \sigma$ and a vertex $v\in \sigma$ with $v<u$, for every $u\in \tau$. Then $v=v_k$ for some $0\leq k\leq n$, and $\tau\subset \tau_{k+1}$. If $z\in |\tau|\subset |\tau_{k+1}|$ and $t\in [0,1]$, it follows from that $$h\big(tv+(1-t)z\big)=h\big(tv_k+(1-t)z\big)=
    H_k(h(z),t)=H_v(h(z),t).\qedhere$$

Let $\Sigma$ be an oriented simplicial complex with respect to some partial order $\leq$ on $\Sigma^0$, and $\Phi,\Psi:\Sigma^0\operatorname{\leadsto}Y$ be as in Theorem \[theorem-Near-sel-v7:1\]. For each vertex ${v\in \Sigma^0}$, fix a contraction $H_v:\Phi(v)\times[0,1]\to \Psi(v)$ into some point ${p_v\in \Psi(v)}$. Since each simplex $\sigma\in \Sigma$ is oriented with respect to $\leq$, by Proposition \[proposition-Near-sel-v8:1\] applied with $\Phi\operatorname{\upharpoonright}\sigma$, $\Psi\operatorname{\upharpoonright}\sigma$ and the fixed contractions $H_v$, $v\in \sigma$, there exists a unique continuous map $h_\sigma:|\sigma|\to Y$ satisfying both properties \[item:Near-sel-v8:1\] and \[item:Near-sel-v8:2\] in that proposition. Suppose that $\sigma_1,\sigma_2\in \Sigma$ are simplices which have a common face $\tau=\sigma_1\cap
  \sigma_2$. Then the linear order $\leq$ on $\tau$ is the same in each one of the oriented simplices $(\sigma_i,\leq)$, $i=1,2$. Hence, by Proposition \[proposition-Near-sel-v8:1\], we get that $h_{\sigma_1}\operatorname{\upharpoonright}|\tau|= h_\tau=h_{\sigma_2}\operatorname{\upharpoonright}|\tau|$. Accordingly, we may define a map $h:|\Sigma|\to Y$ by $h\operatorname{\upharpoonright}|\sigma|=h_\sigma$, $\sigma\in \Sigma$. This map $h$ is as required. Indeed, it is continuous because so is each $h_\sigma$, $\sigma\in \Sigma$. To see that $h$ also has the property in , take a simplex $\sigma\in \Sigma$, and set $v=\min\sigma$. If $\sigma=\{v\}$, then by \[item:Near-sel-v8:1\] of Proposition \[proposition-Near-sel-v8:1\], $h(|\sigma|)=h(v)=h_\sigma(v)=p_v\in \Psi(v)$. Otherwise, if $\tau=\sigma\setminus\{v\}\neq {\varnothing}$, it follows from \[item:Near-sel-v8:2\] of Proposition \[proposition-Near-sel-v8:1\] that for every $z\in |\tau|$ and $t\in [0,1]$, $$h\big(tv+(1-t)z\big)= h_\sigma\big(tv
    +(1-t)z\big)=H_v(h_\sigma(z),t)\in \Psi(v).$$ Accordingly, $h(|\sigma|)\subset \Psi(v)$ because $|\sigma|=|\tau|*v$.

Recall that the $k$-skeleton $\Sigma^k$, $k\geq 0$, of a simplicial complex $\Sigma$ is the simplicial subcomplex $\Sigma^{k}=\{\sigma\in \Sigma:\operatorname{Card}(\sigma)\leq k+1\}$. A simplicial complex $\Sigma$ is called *finite-dimensional* if $\Sigma=\Sigma^k$ for some ${k\geq 0}$, and we say that $\Sigma$ is *$k$-dimensional* if $k\geq0$ is the smallest integer with this property, i.e. for which $\Sigma=\Sigma^k$. If $\Sigma$ is an $(n+1)$-dimensional simplicial complex which is oriented with respect to some partial order $\leq$ on $\Sigma^0$, then we may consider the chain of subcomplexes $\Sigma_{n+1}\subset \Sigma_n\subset \dots \subset \Sigma_0=\Sigma$ defined by $$\label{eq:Near-sel-v9:2}
  \begin{cases}
    \Sigma_k=(\Sigma_k)^{n+1-k}, &\text{and}\\
    \min\sigma\notin (\Sigma_{k+1})^0, &\text{whenever
      $\sigma\in \Sigma_k\setminus \Sigma_{k+1}$.}
  \end{cases}$$ Namely, let $V_0$ be the minimal elements of $\Sigma^0$, i.e. $V_0$ consists of all $v\in \Sigma^0$ such that $v=u$ whenever $u\in
\Sigma^0$ with $u\leq v$. Next, let $V_1$ be the minimal elements of $\Sigma^0\setminus V_0$. Thus, inductively, for each $k\leq n$, we may define the set $V_{k+1}$ which consists of the minimal elements of $\Sigma^0\setminus (V_0\cup\dots\cup V_k)$. Since $\Sigma$ is $(n+1)$-dimensional, each simplex of $\Sigma$ has at most $n+2$ elements. Therefore, $\Sigma^0= V_0\cup \dots \cup V_{n+1}$ and we may now define the required subcomplexes by $$\label{eq:Near-sel-vgg:2}
  \Sigma_k=\left\{\sigma\in \Sigma: \sigma\subset V_k\cup\dots \cup
    V_{n+1}\right\},\quad k\leq n+1.$$

Our second construction deals with finite-dimensional oriented simplicial complexes by relaxing the requirement in Theorem \[theorem-Near-sel-v7:1\] that $\Phi{\lhook\joinrel\xrightarrow{\infty}}\Psi$. To this end, for a space $Y$ and $S\subset T\subset Y$, we will write that $S{\lhook\joinrel\xrightarrow{\omega}} T$ if for each finite-dimensional simplicial complex $\Sigma$, any continuous map $g:|\Sigma|\to S$ is null-homotopic in $T$. Just like before, for mappings $\Phi,\Psi:Z\operatorname{\leadsto}Y$, we will use $\Phi{\lhook\joinrel\xrightarrow{\omega}}\Psi$ to designate that $\Phi(z){\lhook\joinrel\xrightarrow{\omega}} \Psi(z)$, for every $z\in Z$.

For a simplicial complex $\Sigma$ and a point $v$ with $v\notin |\Sigma|$, the *cone* on $\Sigma$ with a vertex $v$, is the simplicial complex defined by $$\label{eq:Near-sel-v9:3}
  \Sigma*v=\Sigma\cup\left\{\sigma\cup \{v\}:
    \sigma\in\Sigma\right\}\cup \{v\}.$$ According to , $|\Sigma*v|$ is the cone $|\Sigma|*v$ over the geometric realisation $|\Sigma|$ of $\Sigma$. Moreover, for a space $T$ and $S\subset T$, we have that $S{\lhook\joinrel\xrightarrow{\omega}} T$ iff for each finite-dimensional simplicial complex $\Sigma$ and $v\notin |\Sigma|$, any continuous map $g:|\Sigma|\to S$ can be extended to a continuous map $h:|\Sigma|*v\to T$ on the cone $|\Sigma|*v=|\Sigma*v|$, compare with Proposition \[proposition-Near-sel-v7:2\]. Thus, we also have the following theorem.

\[theorem-Near-sel-v9:1\] Let $\Sigma$ be a finite-dimensional simplicial complex which is oriented with respect to some partial order $\leq$ on $\Sigma^0$, and $\Phi,\Psi:\Sigma^0\operatorname{\leadsto}Y$ be mappings into a space $Y$ such that $\Phi{\lhook\joinrel\xrightarrow{\omega}} \Psi$ and $\Psi(v)\subset \Phi(u)$, whenever $u,v\in \Sigma^0$ with $u<v$. Then there exists a continuous map $h:|\Sigma|\to Y$ such that $$\label{eq:Near-sel-v9:1}
    h(|\sigma|)\subset \Psi(\min \sigma),\quad\text{for every
      $\sigma\in \Sigma$.}$$

Suppose that $\Sigma$ is $(n+1)$-dimensional, and consider the chain of subcomplexes $\Sigma_{n+1}\subset \Sigma_n\subset\dots \subset \Sigma_0=\Sigma$ with the properties in , see also . We are going to construct continuous maps $h_k:|\Sigma_k|\to Y$, $k\leq n+1$, satisfying and such that each $h_k$ is an extension of $h_{k+1}$. Then we can take $h=h_0$ because $\Sigma_0=\Sigma$. To this end, using that $\Sigma_{n+1}=(\Sigma_{n+1})^0$ is $0$-dimensional, we may define $h_{n+1}:\Sigma_{n+1}\to Y$ by $h_{n+1}(v)\in \Psi(v)$, for every $v\in \Sigma_{n+1}$. Evidently, this $h_{n+1}$ satisfies with respect to any singleton (simplex) of $\Sigma_{n+1}$.

Suppose that $h_{k+1}:\big|\Sigma_{k+1}\big|\to Y$ is a continuous map satisfying , and let us show how to extend it to a continuous map $h_{k}:\big|\Sigma_{k}\big|\to Y$ preserving this property. So, take any vertex $v\in \Sigma^0$ with $\{v\}\in \Sigma_k\setminus \Sigma_{k+1}$, and consider the cone $\Omega_v* v\subset \Sigma_k$ associated to the subcomplex $\Omega_v=\big\{\sigma\in \Sigma_{k+1}: \sigma\cup\{v\}\in
  \Sigma_{k}\big\}$, see . If $\Omega_v={\varnothing}$, then $\Omega_v*v$ is the singleton $\{v\}$, and we define $h_v:|\Omega_v*v|\to Y$ by $h_v(v)\in
  \Psi(v)$. Suppose that $\Omega_v\neq {\varnothing}$, and take any simplex $\sigma\in \Omega_v\subset \Sigma_{k+1}$. According to , $v<\min\sigma$ and, therefore, $\Psi(\min\sigma)\subset \Phi(v)$. Since $h_{k+1}$ satisfies , this implies that $h_{k+1}(|\sigma|)\subset \Phi(v)$. Thus, $h_{k+1}(|\Omega_v|)\subset \Phi(v)$ and using that $\Phi(v){\lhook\joinrel\xrightarrow{\omega}} \Psi(v)$, there exists a continuous extension $h_v:|\Omega_v*v|\to \Psi(v)$ of $h_{k+1}\operatorname{\upharpoonright}|\Omega_v|$. Evidently, $h_v$ satisfies with respect to the simplicial complex $\Omega_v*v$. Finally, we may define the required continuous extension $h_k:|\Sigma_k|\to Y$ of $h_{k+1}$ by letting $h_k\operatorname{\upharpoonright}|\Sigma_{k+1}|=h_{k+1}$ and $h_k\operatorname{\upharpoonright}|\Omega_v*v|=h_v$, for every vertex $v\in \Sigma^0$ with $\{v\}\in \Sigma_k\setminus \Sigma_{k+1}$.

\[remark-Near-sel-v10:1\] The embedding relation ${\lhook\joinrel\xrightarrow{\omega}}$ in Theorem \[theorem-Near-sel-v9:1\] was used only with subcomplexes of $\Sigma$. Hence, this theorem remains valid if the property “$\Phi{\lhook\joinrel\xrightarrow{\omega}} \Psi$” is relaxed to the one that for each simplicial subcomplex $\Omega\subset \Sigma$ and $v\in \Sigma^0$, any continuous map $g:|\Omega|\to \Phi(v)$ is null-homotopic in $\Psi(v)$.

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

If a paracompact space is a countable union of closed $C$-subspaces, then it is itself a $C$-space [@gutev-valov:00 Theorem 4.1], see also [@addis-gresham:78 Theorem 2.7]. Accordingly, the space $X$ in Theorem \[theorem-Selections-IDS:1\] is a $C$-space. In this section, we show that Theorem \[theorem-Selections-IDS:1\] remains valid for all paracompact $C$-spaces. It is based on the relation “$S{\lhook\joinrel\xrightarrow{\infty}}T$” defined in the previous section, which is clearly a monotone embedding relation. In its terms, a subset $S$ of a metric space $(Y,\rho)$ is *CE* iff it is *uniformly $UV^\infty$* (see Section \[sec:monot-embedd-relat\]), namely if for every $\varepsilon>0$ there exists $\delta\in (0,\varepsilon]$ such that $\mathbf{O}_\delta(S){\lhook\joinrel\xrightarrow{\infty}} \mathbf{O}_\varepsilon(S)$.

\[theorem-Finite-C-v6:1\] Let $X$ be a paracompact $C$-space, $(Y,\rho)$ be a metric space and $\varphi:X\operatorname{\leadsto}Y$ be a $\rho$-continuous mapping such that each $\varphi(x)$, $x\in X$, is uniformly $UV^\infty$. Then $\varphi$ has a continuous $\varepsilon$-selection, for every continuous function ${\varepsilon:X\to (0,+\infty)}$.

For a continuous function $\varepsilon:X\to (0,+\infty)$, take open covers $\mathscr{U}_n$, $n<\omega$, of $X$ and mappings $\Phi_n,\Psi_n:\mathscr{U}_n\operatorname{\leadsto}Y$ as those in Theorem \[theorem-Near-sel-v1:1\] (applied with the relation ${\lhook\joinrel\xrightarrow{\infty}}$ and the function $\varepsilon$). Then, since $X$ is a $C$-space, there exists a sequence $\mathscr{V}_{n}$, $n<\omega$, of pairwise disjoint families of nonempty open subsets of $X$ such that each $\mathscr{V}_n$, $n<\omega$, refines $\mathscr{U}_n$, and $\mathscr{V}=\bigcup_{n<\omega}\mathscr{V}_n$ is a cover for $X$. Moreover, we assume that the families $\mathscr{V}_n$, $n<\omega$, have no common elements.

Let $\mathscr{N}(\mathscr{V})$ be the *nerve* of $\mathscr{V}$, i.e. the simplicial complex defined by $$\label{eq:Near-sel-v10:1}
    \mathscr{N}(\mathscr{V})= \left\{\sigma\subset \mathscr{V}:\sigma\
      \text{is finite and}\ \bigcap\sigma\neq{\varnothing}\right\}.$$ Then there exists a natural orientation of $\mathscr{N}(\mathscr{V})$ generated by the families $\mathscr{V}_n$, $n<\omega$. Namely, define a partial order $\leq$ on $\mathscr{V}$ by $V<W$, whenever $V\in \mathscr{V}_n$ and $W\in \mathscr{V}_k$ are such that $V\cap W\neq {\varnothing}$ and $n<k$. Evidently, by , this partial order makes $\mathscr{N}(\mathscr{V})$ an oriented simplicial complex.

Since each $\mathscr{V}_n$ refines $\mathscr{U}_n$, there are maps $r_n:\mathscr{V}_n\to \mathscr{U}_n$, $n<\omega$, such that $V\subset r_n(V)$, for every $V\in \mathscr{V}_n$. We may now define mappings $\Phi,\Psi:\mathscr{V}\operatorname{\leadsto}Y$ by $\Phi\operatorname{\upharpoonright}\mathscr{V}_n=\Phi_n\circ r_n$ and $\Psi\operatorname{\upharpoonright}\mathscr{V}_n=\Psi_n\circ r_n$, $n<\omega$. According to Theorem \[theorem-Near-sel-v1:1\] and the definition of the partial order $\leq$ on $\mathscr{V}$, for every $V\in \mathscr{V}$ we have that $$\label{eq:Near-sel-v4:2}
    \begin{cases}
      \Phi(V){\lhook\joinrel\xrightarrow{\infty}}\Psi(V)\subset
      \mathbf{O}_{\varepsilon(p)}(\varphi(p)),\
      \text{whenever  $p\in V$, and }\\
      \Psi(V)\subset \Phi(U),\ \text{whenever $U\in \mathscr{V}$ with
        $U<V$ and $U\cap V\neq {\varnothing}$.}
    \end{cases}$$ Thus, by Theorem \[theorem-Near-sel-v7:1\], there exists a continuous map ${h:\big|\mathscr{N}(\mathscr{V})\big|\to Y}$ satisfying with respect to this orientation of $\mathscr{N}(\mathscr{V})$. Finally, since $X$ is paracompact, the cover $\mathscr{V}$ has a canonical map $g:X\to |\mathscr{N}(\mathscr{V})|$. In other words, 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 for any sequence $\{\mathscr{U} _n:n<\omega\}$ of open covers of $X$ there exists a finite sequence $\{\mathscr{V} _n:n\leq k\}$ of discrete 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 $X$.

Regarding the proper place of finite $C$-spaces, it was shown by Valov in [@valov:00 Proposition 2.2] that a space $X$ is a finite $C$-space if and only if its Čech-Stone compactification $\beta X$ is a $C$-space. Accordingly, each compact $C$-space is a finite $C$-space. On the other hand, there are $C$-spaces whose Čech-Stone compactification is not a $C$-space, see e.g.[@MR2080284 Remark 3.7].

In this section, we extend Theorem \[theorem-Selections-IDS-v4:1\] to all paracompact finite $C$-spaces. It is based on the relation “$S{\lhook\joinrel\xrightarrow{\omega}} T$” defined in Section \[sec:orient-simpl-compl\], which is another example of a monotone embedding relation. In terms of this relation, we shall say that a subset $S$ of a metric space $(Y,\rho)$ is *uniformly $UV^\omega$* if for each $\varepsilon>0$ there exists $\delta\in (0,\varepsilon]$ such that $\mathbf{O}_\delta(S){\lhook\joinrel\xrightarrow{\omega}} \mathbf{O}_\varepsilon(S)$.

\[theorem-Finite-C-v5:1\] Let $X$ be a paracompact finite $C$-space, $(Y,\rho)$ be a metric space and $\varphi:X\operatorname{\leadsto}Y$ be a $\rho$-continuous mapping such that each $\varphi(x)$, $x\in X$, is uniformly $UV^\omega$. Then $\varphi$ has a continuous $\varepsilon$-selection, for every continuous function ${\varepsilon:X\to (0,+\infty)}$.

We proceed as in the proof of Theorem \[theorem-Finite-C-v6:1\], but now using Theorem \[theorem-Near-sel-v9:1\] instead of Theorem \[theorem-Near-sel-v7:1\]. Namely, for a continuous function $\varepsilon:X\to (0,+\infty)$, take open covers $\mathscr{U}_n$, $n<\omega$, of $X$ and mappings $\Phi_n,\Psi_n:\mathscr{U}_n\operatorname{\leadsto}Y$ as those in Theorem \[theorem-Near-sel-v1:1\] (applied with the embedding relation ${\lhook\joinrel\xrightarrow{\omega}}$ and the function $\varepsilon$). Since $X$ is a paracompact finite $C$-space, by Theorem \[theorem-Selections-IDS-v2:2\], there exists a finite sequence $\mathscr{V}_{n}$, $n\leq k$, of discrete families of nonempty open subsets of $X$ such that each $\mathscr{V}_n$, $n\leq k$, refines $\mathscr{U}_n$, and $\mathscr{V}=\bigcup_{n\leq k}\mathscr{V}_n$ is a cover for $X$. Next, precisely as in the proof of Theorem \[theorem-Finite-C-v6:1\], we define a partial order on $\mathscr{V}$ and mappings $\Phi,\Psi:\mathscr{V}\operatorname{\leadsto}Y$ such that for every $V\in \mathscr{V}$, $$\label{eq:Near-sel-v10:2}
    \begin{cases}
      \Phi(V){\lhook\joinrel\xrightarrow{\omega}}\Psi(V)\subset
      \mathbf{O}_{\varepsilon(p)}(\varphi(p)),\
      \text{whenever  $p\in V$, and }\\
      \Psi(V)\subset \Phi(U),\ \text{whenever $U\in \mathscr{V}$ with
        $U<V$ and $U\cap V\neq {\varnothing}$.}
    \end{cases}$$ Thus, by Theorem \[theorem-Near-sel-v9:1\], there exists a continuous map $h:|\mathscr{N}(\mathscr{V})|\to Y$ satisfying with respect to the oriented simplicial complex $\mathscr{N}(\mathscr{V})$.

We may now finish the proof precisely as before. Namely, since $X$ is paracompact, the cover $\mathscr{V}$ admits a continuous map $g:X\to |\mathscr{N}(\mathscr{V})|$ such that $g(x)\in |\sigma_x|$, $x\in X$, where $\sigma_x=\{V\in \mathscr{V}: x\in V\}\in \mathscr{N}(\mathscr{V})$, see [@dowker:47]. Then the composite map $f=h\circ g$ is a continuous $\varepsilon$-selection for $\varphi$. Indeed, $x\in \min \sigma_x$ and by and , we get that $f(x)=h(g(x))\in h(|\sigma_x|)\subset \Psi(\min \sigma_x)\subset
  \mathbf{O}_{\varepsilon(x)}(\varphi(x))$.

\[remark-Near-sel-vgg:2\] For an infinite cardinal $\tau$, we shall say that a subset $S$ of a metric space $(Y,\rho)$ is *uniformly $\tau$-$UV^\omega$* if for each $\varepsilon>0$ there exists $\delta\in(0,\varepsilon]$ such that for every finite-dimensional simplicial complex $\Sigma$ with $\operatorname{Card}(\Sigma^0)<\tau$, every continuous map $g:|\Sigma|\to \mathbf{O}_\delta(S)$ is null-homotopic in $\mathbf{O}_\varepsilon(S)$. Also, let us recall that the *degree of compactness* of space $X$ is the least infinite cardinal $\kappa(X)$ such that every open cover of $X$ has an open refinement of cardinality less than ${\kappa(X)}$, see [@choban-mihaylova-nedev:08]. Using the same proof, Theorem \[theorem-Finite-C-v5:1\] remains valid if each $\varphi(x)$, $x\in X$, is only assumed to be uniformly $\kappa(X)$-$UV^\omega$. Indeed, if $\mathscr{V}_n$, $n\leq k$, are as in that proof, then ${\operatorname{Card}(\mathscr{V})<\kappa(X)}$ because each family $\mathscr{V}_n$, $n\leq k$, is discrete. Hence, according to Remark \[remark-Near-sel-v10:1\], we can still apply Theorem \[theorem-Finite-C-v6:1\] to get a continuous map $h:|\mathscr{N}(\mathscr{V})|\to Y$ satisfying with respect to the oriented simplicial complex $\mathscr{N}(\mathscr{V})$. Having already the map $h$, we can finish the proof precisely as before. One benefit of this refined version of Theorem \[theorem-Finite-C-v5:1\] is that it implies Theorem \[theorem-Selections-IDS-v4:1\]. Namely, for a compact space $X$ and a metric space $(Y,\rho)$, a mapping $\varphi:X\operatorname{\leadsto}Y$ has uniformly $\kappa(X)$-$UV^\omega$-values precisely when its values are as in Theorem \[theorem-Selections-IDS-v4:1\].

Approximate Selections and Absolute Retracts
============================================

Here, by an A(N)R we mean a metrizable space which is an *Absolute* (*Neighbourhood*) *Retract* for the metrizable spaces. A closed subset $Y\subset E$ of a metric space $(E,\rho)$ is a *uniform neighbourhood retract* of $E$ if to every $\varepsilon > 0$ there corresponds $\eta(\varepsilon) > 0$ such that there exists a retraction $r:\mathbf{O}_{\eta(+\infty)}(Y)=
\bigcup_{\varepsilon>0}\mathbf{O}_{\eta(\varepsilon)}(Y)\to Y$ with $\rho(z,r(z)) < \varepsilon$, whenever $z\in \mathbf{O}_{\eta(\varepsilon)}(Y)$. If one can take $\eta(+\infty) = +\infty$ (so that the domain of $r$ is always $E$), then $Y$ is called a *uniform retract* of $E$. A metric space $(Y,\rho)$ is called a *uniform* A(N)R if it is a uniform (neighbourhood) retract of any metric space containing $(Y,\rho)$ isometrically as a closed subset. Uniform ANR’s and AR’s were considered by Toruńczyk [@MR365471; @MR445503] and Michael [@michael:79], see also Dugundji [@dugundji:58]. It should be remarked that these uniform retracts are slightly different from those considered by Isbell [@MR141074; @isbell:64]. We now have the following application of Theorem \[theorem-Finite-C-v6:1\].

\[corollary-Near-sel-vgg:1\] Let $X$ be a paracompact $C$-space, $(Y,\rho)$ be a metric space which is a uniform ANR, and $\varphi:X\to \mathscr{F}(Y)$ be a $\rho$-continuous mapping such that each $\varphi(x)$, $x\in X$, is a uniform retract of $Y$. Then $\varphi$ has a continuous $\varepsilon$-selection, for every continuous function ${\varepsilon:X\to (0,+\infty)}$.

According to Theorem \[theorem-Finite-C-v6:1\], it suffices to show that each $\varphi(x)$, $x\in X$, is uniformly $UV^\infty$. To this end, take a point $p\in X$, and let $r:\mathbf{O}_{\eta(+\infty)}(\varphi(p))\to \varphi(p)$ be the corresponding retraction, where $\eta(\varepsilon)$ is as in the definition of a uniform retract for this subset $\varphi(p)\subset Y$. Moreover, since $(Y,\rho)$ is a uniform ANR, to each $\varepsilon>0$ there exists $\gamma(\varepsilon)>0$ such that any two $\gamma(\varepsilon)$-close continuous maps from an arbitrary space to $Y$ are $\varepsilon$-homotopic, see e.g. [@MR3099433 Theorem 6.8.6]. Finally, set $\delta(\varepsilon)= \eta(\gamma(\varepsilon))$, and let us show that this works. So, let $g=r\operatorname{\upharpoonright}\mathbf{O}_{\delta(\varepsilon)}(\varphi(p))$ and $f:\mathbf{O}_{\delta(\varepsilon)}(\varphi(p))\to
  \mathbf{O}_{\delta(\varepsilon)}(\varphi(p))$ be the identity of $\mathbf{O}_{\delta(\varepsilon)}(\varphi(p))$. Since $\delta(\varepsilon)=\eta(\gamma(\varepsilon))$, we have that $\rho(f(z),g(z))= \rho(z,r(z))< \gamma(\varepsilon)$, for every $z\in \mathbf{O}_{\delta(\varepsilon)}(\varphi(p))$. Hence, $f$ and $g$ are $\gamma(\varepsilon)$-close and by the choice of $\gamma(\varepsilon)$, they are also $\varepsilon$-homotopic. On the other hand, $\varphi(p)$ is contractible being an AR, see [@dugundji:58 Theorem 12.4]. Therefore, $g$ is null-homotopic in $\varphi(p)$. Thus, the map $f$, i.e. the identity of $\mathbf{O}_{\varepsilon}(\varphi(p))$, is null-homotopic as well.

In case the domain of $\varphi$ is a paracompact finite $C$-space, we may relax both properties that each $\varphi(x)$, $x\in X$, is a uniform ANR and a contractible set (being an AR). In fact, the contractibility can be also relaxed in Corollary \[corollary-Near-sel-vgg:1\]. Namely, for a subset $S\subset T$ of a space $T$ and $k\geq 0$, we write that $S{\lhook\joinrel\xrightarrow{k}} T$ if every continuous map of the $k$-sphere in $S$ can be extended to a continuous map of the $(k+1)$-ball in $T$. A space $S$ is called *aspherical*, or $C^\omega$, if $S{\lhook\joinrel\xrightarrow{k}} S$, for every $k\geq 0$. The following property describing aspherical spaces follows easily from the defining relation that $S{\lhook\joinrel\xrightarrow{k}}S$, for all $k\geq 0$.

\[proposition-Finite-C-v2:1\] A space $S$ is $C^\omega$ if and only if for every simplicial complex $\Sigma$, each continuous map $g:|\Sigma|\to S$ is null-homotopic.

The importance of aspherical spaces is that a metrizable space is an AR iff it is an aspherical ANR, [@dugundji:58 Theorem 12.4].

Evidently, a compact metric space is a uniform A(N)R iff it is an A(N)R. Following Lefschetz [@MR0007094], a compact metric space $(Y,\rho)$ is called $LC^*$ if for every $\varepsilon>0$ there exists $\eta(\varepsilon)>0$ such that if $\Sigma$ is a finite simplicial complex and $K\subset \Sigma$ is a subcomplex containing the vertex set $\Sigma^0$, then every continuous map $g:|K|\to Y$ with $\operatorname{diam}(g(|K\cap \sigma|))<\eta(\varepsilon)$ for every $\sigma\in \Sigma$, can be extended to a continuous map $f:|\Sigma|\to Y$ such that $\operatorname{diam}(f(|\sigma|))<\varepsilon$, $\sigma\in \Sigma$. It was shown by Lefschetz [@MR0007094 Theorem 6.6] that a compact metric space $(Y,\rho)$ is a (uniform) ANR if and only if it is $LC^*$, while $(Y,\rho)$ is an AR iff it is both $LC^*$ and $C^\omega$. In [@michael:79 Theorem 7.1], Michael extended this characterisation to arbitrary uniform ANR’s by showing that a metric space $(Y,\rho)$ is a uniform ANR if and only if it has the above Lefschetz property for arbitrary simplicial complexes $\Sigma$ rather than finite. Here, we are interested in another extension of the $LC^*$ property to arbitrary subsets of metric spaces. Namely, we shall say that a subset $S\subset Y$ of a metric space $(Y,\rho)$ is *uniformly $LC^*$* if for every $\varepsilon>0$ there exists $\eta(\varepsilon)>0$ such that if $\Sigma$ is a finite-dimensional simplicial complex and $K\subset \Sigma$ is a subcomplex containing the vertex set $\Sigma^0$, then each continuous map $g:|K|\to S$ with $\operatorname{diam}(g(|K\cap \sigma|))<\eta(\varepsilon)$ for every $\sigma\in \Sigma$, can be extended to a continuous map $f:|\Sigma|\to S$ such that $\operatorname{diam}(f(|\sigma|))<\varepsilon$, $\sigma\in \Sigma$. The following property of uniformly $LC^*$ sets was essentially established by Michael in [@michael:56b Lemma 11.1].

\[proposition-Near-sel-vgg:1\] Let $S\subset Y$ be a uniformly $LC^*$ subset of a metric space $(Y,\rho)$. Then for every $\varepsilon>0$ there exists $\gamma(\varepsilon)>0$ such that if $Z$ is a finite-dimensional paracompact space and $g:Z\to \mathbf{O}_{\gamma(\varepsilon)}(S)$ is a continuous map, then there exists a continuous map $h:Z\to S$ such that $h(z)\in \mathbf{O}_\varepsilon(g(z))$, for every $z\in Z$.

We repeat the proof of [@michael:56b Lemma 11.1]. Namely, let $\gamma(\varepsilon)=\frac14\eta\left(\frac{2\varepsilon}3\right)$, where $\eta(\varepsilon)\leq \varepsilon$ is as in the definition of uniformly $LC^*$ of $S$. Take a finite-dimensional paracompact space $Z$ and a continuous map $g:Z\to \mathbf{O}_{\gamma(\varepsilon)}(S)$. Then there exists a locally finite open cover $\mathscr{U}$ of $Z$ such that its nerve $\mathscr{N}(\mathscr{U})$ is finite-dimensional (see [@dowker:47 Theorem 3.5]) and $\operatorname{diam}(g(U))<\gamma(\varepsilon)$, for every $U\in \mathscr{U}$. Next, identifying the $0$-skeleton of $\mathscr{N}(\mathscr{U})$ with $\mathscr{U}$, define a (continuous) map $\ell:\mathscr{U}\to S$ such that $\ell(U)\in \mathbf{O}_{\gamma(\varepsilon)}(g(U))$, for every $U\in \mathscr{U}$. Hence, we also have that $$\label{eq:Near-sel-vgg:5}
    \rho(\ell(U),g(z))<2\gamma(\varepsilon)\leq
    \frac\varepsilon3,\quad \text{for every 
      $z\in U\in \mathscr{U}$.}$$ Thus, for $U,V\in \mathscr{U}$ with $U\cap V\neq {\varnothing}$, we get that $\rho(\ell(U),\ell(V))<4\gamma(\varepsilon)=
  \eta\left(\frac{2\varepsilon}3\right)$. Since $\mathscr{N}(\mathscr{U})$ is finite-dimensional and $\eta\left(\frac{2\varepsilon}3\right)$ is as in the definition of uniformly $LC^*$ of $S$, the map $\ell$ can be extended to a continuous map $f:|\mathscr{N}(\mathscr{U})|\to S$ such that ${\operatorname{diam}(f(|\sigma|))<\frac{2\varepsilon}3}$, for every $\sigma\in \mathscr{N}(\mathscr{U})$. Finally, let $\pi:Z\to |\mathscr{N}(\mathscr{U})|$ be a canonical map and $h:Z\to S$ be the composite map $h=f\circ \pi$. If $z\in Z$, then $z\in \bigcap \sigma$ for some $\sigma\in
  \mathscr{N}(\mathscr{U})$. Take any $U\in \sigma\subset
  |\sigma|$. Since $\pi(z),U\in |\sigma|$ and $f(U)=\ell(U)$, it follows from that $$\rho(h(z),g(z))\leq \rho(f(\pi(z)),f(U))+ \rho(\ell(U),g(z))
                     <\frac{2\varepsilon}3 +\frac{\varepsilon}3
                     =\varepsilon.\qedhere$$

We may now apply Theorem \[theorem-Finite-C-v5:1\] to get the following further consequence.

\[corollary-Near-sel-vgg:2\] Let $(Y,\rho)$ be a metric space which is a uniform ANR, $X$ be a paracompact finite $C$-space and $\varphi:X\operatorname{\leadsto}Y$ be a $\rho$-continuous mapping such that each $\varphi(x)$, $x\in X$, is both $C^\omega$ and uniformly $LC^*$. Then $\varphi$ has a continuous $\varepsilon$-selection, for every continuous function $\varepsilon:X\to (0,+\infty)$.

According to Theorem \[theorem-Finite-C-v5:1\], it suffices to show that each $\varphi(x)$, $x\in X$, is uniformly $UV^\omega$. To this end, take a point $p\in X$, and let $\gamma(\varepsilon)$ be as in Proposition \[proposition-Near-sel-vgg:1\]. Since $(Y,\rho)$ is a uniform ANR, as in the proof of Corollary \[corollary-Near-sel-vgg:1\], to each $\varepsilon>0$ there exists $\eta(\varepsilon)>0$ such that any two $\eta(\varepsilon)$-close continuous maps from an arbitrary space to $Y$ are $\varepsilon$-homotopic. Finally, take $\delta(\varepsilon)= \gamma(\eta(\varepsilon))$, and let us show that this works. So, let $g:|\Sigma|\to \mathbf{O}_{\delta(\varepsilon)}(\varphi(p))$ be a continuous map for some finite-dimensional simplicial complex $\Sigma$. Then by Proposition \[proposition-Near-sel-vgg:1\], there exists a continuous map ${h:|\Sigma|\to \varphi(p)}$ such that $\rho(h(z),g(z))<\eta(\varepsilon)$, for all $z\in
  |\Sigma|$. Accordingly, $g$ and $h$ are $\eta(\varepsilon)$-close and by the choice of $\eta(\varepsilon)$, the maps $g$ and $h$ are $\varepsilon$-homotopic. On the other hand, by Proposition \[proposition-Finite-C-v2:1\], $h:|\Sigma|\to \varphi(p)$ is null-homotopic in $\varphi(p)$ because $\varphi(p)$ is aspherical. This implies that $g$ is null-homotopic in $\mathbf{O}_{\varepsilon}(\varphi(p))$, which shows that $\varphi(p)$ is uniformly $UV^\omega$.

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