Next, define $\Phi_n,\Psi_n:\mathscr{U}_n\operatorname{\leadsto}Y$ by $$\label{eq:Near-sel-v5:1}
    \Phi_n(U)=\mathbf{O}_{\delta_n(U)}(\varphi(\pi_n(U)))\quad
    \text{and}\quad
    \Psi_n(U)=\mathbf{O}_{\varepsilon_n(U)}(\varphi(\pi_n(U))),\ U\in
    \mathscr{U}_n.$$

To see \[item:Near-sel-v3:2\], take $k>n$, $U\in \mathscr{U}_n$ and $V_k\in \mathscr{U}_{k}$ with $V_k\cap U\neq {\varnothing}$. Since each $\mathscr{U}_{i+1}$ refines $\mathscr{U}_i$, $i<\omega$, there are elements $V_i\in \mathscr{U}_i$, $n+1\leq i< k$, such that $V_{i+1}\subset V_{i}$ for every $n+1\leq i< k$. Take a point $q\in V_k\cap U\subset V_{n+1}\cap U$. Since $\pi_k(V_k),q\in V_{n+1}$ and $q,\pi_n(U)\in U$, it follows from that $$\begin{aligned}
    H(\rho)(\varphi(\pi_{k}(V_k)),
    &\varphi(\pi_n(U)))\leq\\
    &\leq H(\rho)(\varphi(\pi_{k}(V_k)),\varphi(q))+
      H(\rho)(\varphi(q),\varphi(\pi_n(U)))\\
    &<
      \frac{\delta_{n+1}(V_{n+1})}2+\frac{\delta_n(U)}2
      \leq
      \frac12 \frac{\delta_{n}(U)}3+ \frac{\delta_{n}(U)}2=
      \frac{2\delta_{n}(U)}3.  
  \end{aligned}$$ For the same reason, $\varepsilon_k(V_k)\leq \dots \leq \varepsilon_{n+1}(V_{n+1})\leq
  \frac{\delta_n(U)}3$. Accordingly, $$\begin{aligned}
    \Psi_{k}(V_k)=\mathbf{O}_{\varepsilon_{k}(V_k)}(\varphi(\pi_{k}(V_k)))
    &\subset
      \mathbf{O}_{\frac{\delta_n(U)}3}(\varphi(\pi_{k}(V_{k})))\\
    &\subset
      \mathbf{O}_{\frac{\delta_n(U)}3+
      \frac{2\delta_n(U)}3}(\varphi(\pi_n(U)))=\Phi_n(U).
  \end{aligned}$$

Finally, we show that \[item:Near-sel-v3:1\] holds as well. The “${\lhook\joinrel\xrightarrow{*}}$”-embedding property in \[item:Near-sel-v3:1\] is evident from and . As for the inclusion $\Psi_n(U)\subset \mathbf{O}_{\varepsilon(p)}(\varphi(p))$, where ${p\in U\in \mathscr{U}_n}$, it follows from the special case of $n=0$. Indeed, for $n>0$, take $U_0\in \mathscr{U}_0$ with $U\subset U_0$, which is possible because $\mathscr{U}_n$ refines $\mathscr{U}_0$. Then according to \[item:Near-sel-v3:2\], $\Psi_n(U)\subset \Phi_0(U_0)$ because $U\cap U_0\neq
  {\varnothing}$. Thus, by and , we also have that $$\begin{aligned}
    \Psi_n(U)\subset
    \Phi_0(U_0)&=
                 \mathbf{O}_{\delta_0(U_0)}(\varphi(\pi_0(U_0)))\\
               &\subset \mathbf{O}_{2\delta_0(U_0)}(\varphi(p))\subset
                 \mathbf{O}_{2\varepsilon_0(U_0)}(\varphi(p))\subset 
                 \mathbf{O}_{\varepsilon(p)}(\varphi(p)).\qedhere 
  \end{aligned}$$

Oriented Simplicial Complexes {#sec:orient-simpl-compl}
=============================

As stated in the Introduction, we identify the vertices of a simplicial complex $\Sigma$ with its $0$-skeleton $\Sigma^0$. A 