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\ 