with first $m-1$ elements removed. This changes the metric on the limit, but not the topology, hence $X'$ is homeomorphic to $\pi_m^{-1}(A)$.

The other conditions are trivial to verify.

\[thm:ane limit of polyhedra\] Let $$X = \lim_{\longleftarrow} \left(K_1 \xleftarrow{p_1} K_2 \xleftarrow{p_2} \cdots\right).$$ Assume that for each $i$ the following conditions are satisfied:

1.  $K_i$ is a locally finite-dimensional polyhedron; and

2.  $p_i$ is a quasi-simplicial map that is surjective and $n$-regular.

Then

1.  $X$ is an $ANE(n)$;

2.  each short projection $\pi^k_i \colon K_k \to K_i$ and each long projection $\pi_i \colon X \to X_i$ is a weak $n$-homotopy equivalence;

3.  for each $i$, the covers ${{\mathcal{O}_{i}}}$ and ${{\mathcal{B}_{i}}}$ are $AE(n)$-covers of $X$.

Fix a metric of scale $2^{-i}$ on $K_i$ (see Definition \[def:polyhedron metric\]). Let $(*)$ denote the inverse limit $$X = \lim_{\longleftarrow} \left( K_1 \xleftarrow{p_1} K_2 \xleftarrow{p_2} \cdots\right)$$ along with a sequence ${{\mathcal{B}_{K_i}}}$ of covers, where ${{\mathcal{B}_{K_i}}}$ is a cover of $K_i$ by barycentric stars of vertices.

We will verify that $(*)$ satisfies conditions (A)–(E) of Definition \[def:limit\].

Condition (A): By Lemma \[lem:complex is ane\], every $K_i$ is a complete metric space.

Condition (B): By condition (II), $p_i$ is surjective. By the choice of scale on $K_i$ and 