Would it make sense to define a filtered open covering $$\{U_i\} = U_1 < U_2 < \cdots < U_n = X$$ on a topological space $X$ in order to compute Čech homology? Or does this defeat the purpose?
Context: I have a cosheaf $F$ on a space $X$ in which the stalk is only non trivial for a set of points lying on a vertical line. The only way to define an open covering in which most of the intersections are non-zero is to take the filtered covering described. The aim is to compute Čech homology with coefficents in $F$ given that open covering of $X$.