Homeomorphisms between finite powers of topological spaces

Abstract Let λ be an infinite cardinal number. It is proved that, for each positive integer r , there exists a compact connected homogeneous topologicǎl space X of weight λ such that X n is homeomorphic to X m iff n ≡ m ( mod r ). The cardinality of the set of homeomorphism classes of compact connected homogeneous spaces with this property is exactly 2 λ . Moreover every completely regular space of weight λ is embeddable in a space of this type.