Before revision:  Applying these results to the restricted Cech complex and using the interleaving relations with the Cech complex (or the Rips complex), we formulate conditions guaranteeing that the target space is homotopy equivalent to the Cech complex (or the Rips complex), in terms of the \mu-reach.
After revision:  Applying these results to the restricted Cech complex and using the interleaving relations with the Cech complex (or the Rips complex), we formulate conditions guaranteeing that the target space is homotopy equivalent to the Cech complex (or the Vietoris-Rips complex), in terms of the \mu-reach.