Free abelian topological groups and Pontryagin duality
It is shown that the Pontryagin duality theorem admits an extension to some free abelian topological groups. A Tychonoff space X s said to be a \(\mu\)-space provided any closed bounded subset of X is compact; X is a \(k_ f\)-space provided any real-valued function f on X is continuous iff f is continuous on each compact subset of X. The main result of the paper is the following statement: if X is a \(\mu\)- and \(k_ f\)-space and \(\dim X=0\), then the natural mapping \(\nu_ X: A(X)\to A(X){\hat{\;}}{\hat{\;}}\) is a topological isomorphism of the free abelian topological group A(X) onto its second dual group \(A(X){\hat{\;}}{\hat{\;}}.\) In particular, \(\nu_ X\) is a topological isomorphism for each paracompact k-space X with dim X\(=0\) (Corollary 2).
scientific article; zbMATH DE number 3966488