Calibers, $$\omega $$ -continuous maps and function spaces

Abstract

The operation of extending functions from \(\scriptstyle X\) to \(\scriptstyle \upsilon X\) is \(\scriptstyle \omega \)-continuous, so it is natural to study \(\scriptstyle \omega \)-continuous maps systematically if we want to find out which properties of \(\scriptstyle C_p(X)\) “lift” to \(\scriptstyle C_p(\upsilon X)\). We study the properties preserved by \(\scriptstyle \omega \)-continuous maps and bijections both in general spaces and in \(\scriptstyle C_p(X)\). We show that \(\scriptstyle \omega \)-continuous maps preserve primary \(\scriptstyle \Sigma \)-property as well as countable compactness. On the other hand, existence of an \(\scriptstyle \omega \)-continuous injection of a space \(\scriptstyle X\) to a second countable space does not imply \(\scriptstyle G_\delta \)-diagonal in \(\scriptstyle X\); however, existence of such an injection for a countably compact \(\scriptstyle X\) implies metrizability of \(\scriptstyle X\). We also establish that \(\scriptstyle \omega \)-continuous injections can destroy caliber \(\scriptstyle \omega _1\) in pseudocompact spaces. In the context of relating the properties of \(\scriptstyle C_p(X)\) and \(\scriptstyle C_p(\upsilon X)\), a countably compact subspace of \(\scriptstyle C_p(X)\) remains countably compact in the topology of \(\scriptstyle C_p(\upsilon X)\); however, compactness, pseudocompactness, Lindelof property and Lindelof \(\scriptstyle \Sigma \)-property can be destroyed by strengthening the topology of \(\scriptstyle C_p(X)\) to obtain the space \(\scriptstyle C_p(\upsilon X)\). We show that Lindelof\(\scriptstyle \Sigma \)-property of \(\scriptstyle C_p(X)\) together with \(\scriptstyle \omega _1\) being a caliber of \(\scriptstyle C_p(X)\) implies that \(\scriptstyle X\) is cosmic.