Anthony W. Hager
The $\sigma$-property in $C(X)$

Comment.Math.Univ.Carolin. 57,2 (2016) 231-239.

Abstract:The $\sigma$-property of a Riesz space (real vector lattice) $B$ is: For each sequence $\{b_{n}\}$ of positive elements of $B$, there is a sequence $\{\lambda_{n}\}$ of positive reals, and $b\in B$, with $\lambda_{n}b_{n}\leq b$ for each $n$. This condition is involved in studies in Riesz spaces of abstract Egoroff-type theorems, and of the countable lifting property. Here, we examine when ``$\sigma$'' obtains for a Riesz space of continuous real-valued functions $C(X)$. A basic result is: For discrete $X$, $C(X)$ has $\sigma$ iff the cardinal $|X|< \mathfrak{b}$, Rothberger's bounding number. Consequences and generalizations use the Lindel\"of number $L(X)$: For a $P$-space $X$, if $L(X)\leq \mathfrak{b}$, then $C(X)$ has $\sigma$. For paracompact $X$, if $C(X)$ has $\sigma$, then $L(X)\leq \mathfrak{b}$, and conversely if $X$ is also locally compact. For metrizable $X$, if $C(X)$ has $\sigma$, then $X$ \textit{is} locally compact.

Keywords: Riesz space; $\sigma$-property; bounding number; $P$-space; paracompact; locally compact

DOI: DOI 10.14712/1213-7243.2015.162
AMS Subject Classification: 03E17 06F20 46A40 54C30 54A25 54D20 54D45 54G10