$\Sigma$. 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 