$C$-spaces and simplicial complexes

Given a class $\mathscr G$ of simplicial complexes $G$, we introduce the notion of a $\mathscr G$-$C$-space. In the definition of a $C$-space, open disjoint families $v_i$ refine coverings $u_i$. The nerves of these families are zero-dimensional complexes. In our definition, the nerve of a family $v_i$ must embed in the complex $G_i$ of the class $\mathscr G$. We give a complete characterization of bicompact $\mathscr G$-$C$-spaces.