The $\sigma$-property in $C(X)$

Hager, Anthony W.

Commentationes Mathematicae Universitatis Carolinae, Tome 57 (2016) no. 2, pp. 231-239

Abstract (VO)

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 Lindelöf 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.

Détail

DOI : 10.14712/1213-7243.2015.162

Classification : 03E17, 06F20, 46A40, 54A25, 54C30, 54D20, 54D45, 54G10
Keywords: Riesz space; $\sigma$-property; bounding number; $P$-space; paracompact; locally compact