The space of adding machines generated by continuous self maps of manifolds

Let M be the Cantor space or an n-dimensional manifold with C(M,M) the set of continuous self-maps of M, and . We prove the following:(1)If α≠∞, then Sα(M) is a nowhere dense subset of M×C(M,M) that contains no isolated points.(2)If α⩽β, then .