$A_i \subset \operatorname{Int}A_{i+1}$. If for each $i$, $A_i$ is $AE(n)$, then $A = \bigcup_i A_i$ is $AE(n)$.

Polyhedra
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For a simplicial complex, the underlying polyhedron has two topologies, the Whitehead (weak) topology and the metric topology. The metric topology is the topology of point-wise convergence of barycentric coordinates [@hu1965 p. 100]. The weak topology is metrizable if and only if the complex is locally finite, and it coincides with the metric topology in this case. Since we work in the metric category with complexes that are not locally finite, [**we always assume the metric topology on simplicial complexes**]{} [@lundell1969; @hu1965].

Let $K$ be a simplicial complex. We let $\tau(K)$ denote the [[**triangulation of $K$**]{}]{} (the set of simplices of $K$). We let $V(K)$ denote the [[**vertex set of $K$**]{}]{}. We let $\beta K$ denote the [[**barycentric subdivision of $K$**]{}]{} (i.e., the same space but with a finer triangulation $\tau(\beta K)$).

\[def:polyhedron metric\] Let $K$ be a simplicial complex. Let $\kappa > 0$. Let $\ell_1(V(K))$ denote the Banach space $\{ x \in \mathbb{R}^{V(K)} \colon \sum_{v \in V(K)} | (x)_v | < \infty \}$, where $(x)_v$ denotes the $v$-th coordinate, equipped with the standard $\| \cdot \|_1$ norm. For $v \in 