\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 