L$ by the formula $$\varPhi(x, t) = t \cdot q(x) + (1-t)\cdot x\text{.}$$ Observe that $\operatorname{supp}\varPhi(x, t) \subset \operatorname{supp}(p(x))$ and $V_K \cap \operatorname{supp}\varPhi(x, t) \neq \emptyset$, hence $\varPhi(x, t) \in \operatorname{ost}_L K$. Note that $\varPhi(x, 1) \in K$ because $K$ is a full subcomplex. Hence $\varPhi$ is a deformation retraction of $\operatorname{ost}_L K$ to $K$.

Observe that $$\operatorname{bst}_L K = \{ x \in L \colon \exists_{v \in V_K} (x)_v \geq \max \{ (x)_w \colon w \in V_L \} \}.$$ Since $\varPhi$ preserves the relation $\exists_{v \in V_K} (x)_v \geq \max \{ (x)_w \colon w \in V_L \}$, $\varPhi$ restricted to $\operatorname{bst}_L K \times [0,1]$ is into $\operatorname{bst}_L K$, hence it is a deformation retraction of $\operatorname{bst}_L K$ to $K$.

\[cor:open star swelling\] If $K$ is a polyhedron and $\mathcal{K} = \{ K_i \}_{i \in I}$ is an $AE(n)$-cover of $K$ by subcomplexes, then the collection $$\operatorname{ost}_{\beta K} \mathcal{K} = \{ \operatorname{ost}_{\beta K} K_i \}_{i \in I}$$ is an open $AE(n)$-cover of $K$. Moreover, $\operatorname{ost}_{\beta K} \mathcal{K}$ and $\mathcal{K}$ are isomorphic covers of $K$.

Observe that if $\{ A_j \}_{j \in J}$ is a collection of subcomplexes of a polyhedron $K$, then $$\bigcap_{j \in J} \operatorname{ost}_{\beta K} A_j = 
       \operatorname{ost}_{\beta K} 