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 