  We present results about the Cantor-Bendixson index of some subspaces of a
uniform family F of finite subsets of natural numbers with respect to the
lexicographic order topology. As a corollary of our results we get that for any
omega-uniform family F the restriction F|M is homeomorphic to F iff M contains
intervals of arbitrary length of consecutive integers. We show the connection
of these results with a topological partition problem of uniform families.
