of sets and use the usual notation, ${{\mathcal{F}_{}}} = \{ F_i \}_{i \in I}$, where $I$ denotes the indexing set.

\[def:stars\] Let $K$ be a polyhedron. Let $L \subset K$ be a subcomplex of $K$. The [[**open star $\operatorname{ost}_K L$ of $L$ in $K$**]{}]{} is the complement of the union of all simplices of $K$ that do not intersect $L$: $$\operatorname{ost}_K L = K \setminus \bigcup \{ \delta \in \tau(K) \colon \delta \cap L = \emptyset \}\text{.}$$ The [[**barycentric star $\operatorname{bst}_K L$ of $L$ in $K$**]{}]{} is the union of all simplices of $\beta K$ that intersect $L$: $$\operatorname{bst}_K L = \bigcup \{ \delta \in \tau(\beta K) \colon \delta \cap L \neq \emptyset \}\text{.}$$

We let $${{\mathcal{O}_{K}}} = \{ \operatorname{ost}_K \{ v \} \colon v \in V(K) \}$$ denote the [[**cover of $K$ by open stars of vertices**]{}]{}. We let $${{\mathcal{B}_{K}}} = \{ \operatorname{bst}_K \{ v \} \colon v \in V(K) \}$$ denote a [[**cover of $K$ by barycentric stars of vertices**]{}]{}.

\[lem:stars\] Let $K$ be a polyhedron. The cover ${{\mathcal{B}_{K}}}$ by barycentric stars of vertices is a closed locally finite $AE(\infty)$-cover of $K$. Moreover, if $K$ is locally finite-dimensional, then ${{\mathcal{B}_{K}}}$ is locally finite-dimensional. The cover ${{\mathcal{O}_{K}}}$ by open 