and $\mathscr{V}=\bigcup_{n<\omega}\mathscr{V}_n$ is a cover for $X$. Moreover, we assume that the families $\mathscr{V}_n$, $n<\omega$, have no common elements.

Let $\mathscr{N}(\mathscr{V})$ be the *nerve* of $\mathscr{V}$, i.e. the simplicial complex defined by $$\label{eq:Near-sel-v10:1}
    \mathscr{N}(\mathscr{V})= \left\{\sigma\subset \mathscr{V}:\sigma\
      \text{is finite and}\ \bigcap\sigma\neq{\varnothing}\right\}.$$ Then there exists a natural orientation of $\mathscr{N}(\mathscr{V})$ generated by the families $\mathscr{V}_n$, $n<\omega$. Namely, define a partial order $\leq$ on $\mathscr{V}$ by $V<W$, whenever $V\in \mathscr{V}_n$ and $W\in \mathscr{V}_k$ are such that $V\cap W\neq {\varnothing}$ and $n<k$. Evidently, by , this partial order makes $\mathscr{N}(\mathscr{V})$ an oriented simplicial complex.

Since each $\mathscr{V}_n$ refines $\mathscr{U}_n$, there are maps $r_n:\mathscr{V}_n\to \mathscr{U}_n$, $n<\omega$, such that $V\subset r_n(V)$, for every $V\in \mathscr{V}_n$. We may now define mappings $\Phi,\Psi:\mathscr{V}\operatorname{\leadsto}Y$ by $\Phi\operatorname{\upharpoonright}\mathscr{V}_n=\Phi_n\circ r_n$ and $\Psi\operatorname{\upharpoonright}\mathscr{V}_n=\Psi_n\circ r_n$, $n<\omega$. According to Theorem \[theorem-Near-sel-v1:1\] and the definition of the partial order $\leq$ on $\mathscr{V}$, for every $V\in \mathscr{V}$ we have that $$\label{eq:Near-sel-v4:2}
    \begin{cases}
      \Phi(V){\lhook\joinrel\xrightarrow{\infty}}\Psi(V)\subset
      \mathbf{O}_{\varepsilon(p)}(\varphi(p)),\
      \text{whenever  $p\in V$, and }\\
      \Psi(V)\subset \Phi(U),\ \text{whenever $U\in \mathscr{V}$ with
        $U<V$ and $U\cap V\neq {\varnothing}$.}
    \end{cases}$$ Thus, by Theorem \[theorem-Near-sel-v7:1\], there exists a continuous map ${h:\big|\mathscr{N}(\mathscr{V})\big|\to Y}$ satisfying with respect to this orientation of $\mathscr{N}(\mathscr{V})$. Finally, since $X$ is paracompact, the cover $\mathscr{V}$ has a canonical map $g:X\to |\mathscr{N}(\mathscr{V})|$. In other words, 