Bounded sets in spaces and topological groups
A subset \(A\) of a Tikhonov space \(X\) is C-compact (or relatively pseudocompact) in \(X\) if for every continuous real-valued function \(f\) on \(X\) the image \(f(A)\) is compact (or, every infinite system of open sets in \(X\) meeting \(A\) has an accumulation point in \(A\), resp.). Both concepts coincide in topological groups and are not finitely productive in the category of Tikhonov spaces. It is proved in the present paper that if \(A\) and \(B\) are C-compact (or relatively pseudocompact) in \(X\) and \(Y\), and one of the spaces \(X,Y\) is a topological group, then \(A\times B\) is C-compact (or relatively pseudocompact, resp.) in \(X\times Y\). From other results: 1.3: If \(A_i\) are C-compact in topological groups \(X_i\) then \(\prod A_i\) is C-compact in \(\prod X_i\). 3.3+3.4: The Hewitt-Nachbin realcompactification of an R-factorizable group is an R-factorizable group.
scientific article; zbMATH DE number 1395045