The Eighth Prague Topological Symposium August 18-24, 1996 Economical University Prague, Czech Republic

Abelian Groups of Integer-Valued Continuous Functions by Haruto Ohta ( )

For a topological space X, C(X, Z) denotes the abelian group of continuous functions from X to the discrete group Z of integers. This talk is a survey of the study of C(X, Z) by K. Eda, S. Kamo and myself. We consider no topology on C(X, Z) , but a homomorphism between these groups becomes continuous under the compact-open topology:

Theorem 1. Let X be a Z -compact spaces. Then, every homomorphism \varphi:C(X, Z)--> C(Y, Z) is continuous with respect to the compact-open topology.

A Z -compact space is a space which is homeomorphic to a closed subspace of a power of Z . In view of Theorem 1, one may think that the theory of C(X, Z) is an analogue of that of C_k(X, R) , the linear topological space of real-valued continuous functions with the compact-open topology. However, there are several essential differences between them. For example, as a corollary of Bergman's theorem, we have:

Theorem 2. For 0-dimensional compact spaces X and Y , the groups C(X, Z) and C(Y, Z) are isomorphic if and only if w(X) = w(Y) .

Another example which shows the difference is:

Theorem 3. Let X be a Z -compact space. Then, A = C(X, Z) is reflexive, i.e., A is naturally isomorphic to Hom(Hom(A, Z ), Z ), if and only if every compact subset of X is of nonmeasurable cardinality and k_ZX = X .

Here, k_ZX is the set X equipped with the coarsest topology for which every Z -valued functions which is continuous continuous on every compact set is continuous.