Nearly realcompact spaces
A Tychonoff space \(X\) is nearly realcompact if \(\beta X\setminus\upsilon X\) is dense in \(\beta X\setminus X\), where \(\beta X\) is the Stone- Čech compactification of \(X\) and \(\upsilon X\) is the Hewitt realcompactification of \(X\). While every realcompact space is obviously nearly realcompact, the converse does not hold. The authors obtain several characterizations for nearly realcompact spaces and also several sufficient conditions for a space to be nearly realcompact. Of basic importance in these results are the notions of local compactness and relative pseudocompactness. (A subset \(A\) of a space \(X\) is relatively pseudocompact in \(X\) if every continuous real-valued function on \(X\) is bounded on \(A\).) It is shown that the nearly realcompactness of a space \(X\) is equivalent to \(X\) containing the locally compact part of \(\upsilon X\) and also is equivalent to all relatively pseudocompact cozero-sets in \(X\) being \(\sigma\)-compact (or realcompact, or nearly realcompact). Among interesting nearly realcompact spaces are discrete spaces (with no measurable cardinal assumptions) and topologically complete spaces. Also, if the absolute of a space is nearly realcompact, so is the space; although the converse fails generally, it does hold if the space is nowhere locally compact. The authors also discuss product, sum, and mapping theorems, and several open questions are posed.
scientific article; zbMATH DE number 149082
scientific article