On calibers for $C_p(X)$

Abstract

We present new results regarding calibers in the function spaces $C_p(X)$. Our main theorem is that $C_p(X)$ is strongly Šanin whenever $X$ is a submetrizable space; this improves an earlier result due to Tkachuk: $C_p(X)$ is Šanin whenever $X$ is a submetrizable space. Moreover, we give sufficient conditions to characterize the calibers of $C_p(X)$ when $X$ is a topological sum, and we calculate the calibers of $C_p(X)$ when $X={\prod }_{\xi <\lambda }{X}_{\xi }$ is a product of non-trivial Tychonoff spaces with $i$-weight $\le \lambda$. Furthermore, we calculate the calibers of $C_p(X)$ when $X$ is an interval of ordinals and when $X$ is the one-point $\lambda$-Lindelöf extension of a discrete space of cardinality $\ge \lambda$. This allows to give examples of compact Hausdorff spaces $Z$ such that $iw(Z)={\kappa }^{+}$ and $C_p(Z^{\kappa })$ does not have caliber $iw(Z)$; and examples of spaces $\{Z_{\alpha }:\alpha <cf(\kappa )\}$ such that $\kappa$ is a caliber for $C_p(Z_{\alpha })$ whenever $\alpha <cf(\kappa )$ but it is not a caliber for $C_p({⨁}_{\alpha <cf(\kappa )}{Z}_{\alpha })$.