  Let E be a metric space. We introduce a notion of connectedness index of E,
which is the Hausdor? dimension of the union of non-trivial connected
components of E. We show that the connectedness index of a fractal cube E is
strictly less than the Hausdor? dimension of E provided that E possesses a
trivial connected component. Hence the connectedness index is a new Lipschitz
invariant. Moreover, we investigate the relation between the connectedness
index and topological Hausdor? dimension.
