Abelian groups of continuous functions and their duals

AbstractLet A∗ = Hom (A, Z) for an Abelian group A, were Z is the group of integers. A∗ is endowed with the topology as a subspace of ZA. Then, for a 0-dimensional space X and an infinite cardinal κ the following are equivalent. (1) There exists a free summand of C(X, Z) of rank κ; (2) there exists a subgroup of C(X, Z)∗ isomorphic to Zκ; (3) there exists a compact subset K of βNX with w(K)⩾κ; (4) there exists a compact subset K of C(X, Z)∗ with w(K)⩾κ. There exist groups A such that A∗ is a subgroup of ZN and A∗ is not isomorphic to A∗∗∗