stars of vertices is an open $AE(\infty)$-cover of $K$.

Let $$X = \lim_{\longleftarrow} \left(   
  K_1\xleftarrow{p_1}K_2\xleftarrow{p_2}\cdots \right)$$ Let $v(K_i)$ denote the set of vertices of $K_i$. We let $${{\mathcal{O}_{K_i}}} = \{ O_v = \operatorname{ost}_{K_i} v \}_{v \in v(K_i)}$$ be the cover of $K_i$ by open stars of vertices of $K_i$ (see Definition \[def:stars\]) and $${{\mathcal{O}_{i}}} = \{ \pi^{-1}_i(\operatorname{ost}_{K_i} v) \}_{v \in v(K_i)}$$ be the cover of $X$ by sets of threads that pass through elements of ${{\mathcal{O}_{K_i}}}$.

Let $$X = \lim_{\longleftarrow} \left(   
  K_1\xleftarrow{p_1}K_2\xleftarrow{p_2}\cdots \right)$$ Let $v(K_i)$ denote the set of vertices of $K_i$. We let $${{\mathcal{B}_{K_i}}} = \{ \operatorname{bst}_{K_i} v \}_{v \in v(K_i)}$$ be the cover of $K_i$ by barycentric stars of vertices of $K_i$ (see Definition \[def:stars\]) and $${{\mathcal{B}_{i}}} = \{ \pi^{-1}_i(\operatorname{bst}_{K_i} v) \}_{v \in v(K_i)}$$ be the cover of $X$ by sets of threads that pass through elements of ${{\mathcal{B}_{K_i}}}$.

We say that a cover ${{\mathcal{F}_{}}} = \{ F_i \}_{i \in I}$ is [[**isomorphic**]{}]{} to a cover ${{\mathcal{G}_{}}} = \{ G_i \}_{i \in I}$ if for each $J \subset I$ we have $$\bigcap_{j \in J} F_j \neq \emptyset \iff 
    \bigcap_{j \in J} G_j \neq \emptyset.$$ Note the identical indexing set of ${{\mathcal{F}_{}}}$ and ${{\mathcal{G}_{}}}$.

Let $p \colon Y \to 