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}
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