Note about Lindel\"of $\Sigma$-spaces $\upsilon X$

Abstract

The paper deals with the following problem: Characterize Tichonov spaces $X$ for which its realcompactification $\upsilon X$ is a Lindel\"of $\Sigma$-space. There are many situations (both in Topology and Functional Analysis) where Lindel\"of $\Sigma$ (even $K$-analytic) -spaces $\upsilon X$ appear. For example (see Introduction) if $E$ is a lcs in the class $\mathfrak{G}$ in sense of Cascales and Orihuela ($\mathfrak{G}$ includes among others $(LM)$-spaces and $(DF)$-spaces), then $\upsilon (E',\sigma(E',E))$ is $K$-analytic and $E$ is web-bounded. This provides a general fact (due to Cascales-Kakol-Saxon): If $E\in\mathfrak{G}$, then $\sigma(E',E)$ is K-analytic iff $\sigma(E',E)$ is Lindel\"of. We prove a corresponding result for spaces $C_{p}(X)$ of continuous real-valued maps on $X$ endowed with the pointwise topology: $\upsilon X$ is a Lindel\"of $\Sigma$-space iff $X$ is strongly web-bounding iff $C_{p}(X)$ is web-bounded. Hence the \textit{weak}$^{*}$ dual of $C_{p}(X)$ is a Lindel\"of $\Sigma$-space iff $C_{p}(X)$ is web-bounded and has countable tightness. Applications are provided. For example, every $E\in\mathfrak{G}$ is covered by a family $\{A_{\alpha}:\alpha\in\Omega\}$ of bounded sets for some non-empty set $\Omega\subset\mathbb{N}^{\mathbb{N}}$. DOI: 10.1017/S000497271100270X