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, Lindelöf property and Lindelöf $\scriptstyle \Sigma$-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 Lindelöf$\scriptstyle \Sigma$-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.