Let 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***.
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