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 $X = \prod_{ξ< λ}X_ξ$ is a product of non-trivial Tychonoff spaces with $i$-weight $\leq λ$. Furthermore, we calculate the calibers of $C_p(X)$ when $X$ is an interval of ordinals and when $X$ is the one-point $λ$-Lindelöf extension of a discrete space of cardinality $\geq λ$. This allows to give examples of compact Hausdorff spaces $Z$ such that $iw(Z)=κ^{+}$ and $C_p(Z^κ)$ does not have caliber $iw(Z)$; and examples of spaces $\{Z_α: α<cf(κ)\}$ such that $κ$ is a caliber for $C_p(Z_α)$ whenever $α<cf(κ)$ but it is not a caliber for $C_p(\bigoplus_{α<cf(κ)} Z_α)$.