Čech complexes for finite sets

Let $\mathcal{F}^{m}=\{A\subset[m]\}$, endowed with the symmetric difference metric and the Cardinality Reverse-Lexicographic order. That way, we compute the homotopy type $\operatorname{Čech}\left(\mathcal{F}_{\preceq a}^m; \frac{r}{2}=1\right)$ by relating it to consecutively adding vertices of $\