and the pull-back $p^{-1}({{\mathcal{B}_{L}}})$ is an $AE(n)$-cover of $K$.

The main tool used in the proof is Theorem \[thm:nerve theorem\], which when applied to a canonical map into the nerve of a cover [@nagorko2007] implies the usual Nerve Theorem.

Nerve Theorem for non locally finite covers
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\[lem:deformation\] If $K$ is a full subpolyhedron of a polyhedron $L$, then

1.  $K$ is a deformation retract of $\operatorname{bst}_L K$; and

2.  $K$ is a deformation retract of $\operatorname{ost}_L K$.

We let $V_L$ denote the set of vertices of $L$ and $V_K \subset V_L$ denote the set of vertices of $K$. We regard $L$ as a subspace of $\ell_1(V_L)$ (see Definition \[def:polyhedron metric\]). Let $p \colon L \to \ell_1(V_L)$ be the map that sends all coordinates that do not belong to $K$ to $0$, defined by the formula $$\left( p(x) \right)_v =
  \left\{
  \begin{array}{ll}
  0 & v \in V_L \setminus V_K \\
  (x)_v & v \in V_K
  \end{array}
  \right..$$

We have $$\operatorname{ost}_L K = \{ x \in L \colon p(x) \neq 0 \}.$$ Let $q \colon \operatorname{ost}_L K \to L$ be defined by the formula $$q(x) = \frac{p(x)}{\| p(x) \|_1}.$$ We define a map $\varPhi \colon \operatorname{ost}_L K \times [0, 1] \to 