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

@article{Hager2016,
abstract = {The $\sigma $-property of a Riesz space (real vector lattice) $B$ is: For each sequence $\lbrace b_\{n\}\rbrace $ of positive elements of $B$, there is a sequence $\lbrace \lambda _\{n\}\rbrace $ of positive reals, and $b\in B$, with $\lambda _\{n\}b_\{n\}\le b$ 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 $L(X)\le \mathfrak \{b\}$, then $C(X)$ has $\sigma $. For paracompact $X$, if $C(X)$ has $\sigma $, then $L(X)\le \mathfrak \{b\}$, and conversely if $X$ is also locally compact. For metrizable $X$, if $C(X)$ has $\sigma $, then $X$is locally compact.},
author = {Hager, Anthony W.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Riesz space; $\sigma $-property; bounding number; $P$-space; paracompact; locally compact},
language = {eng},
number = {2},
pages = {231-239},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {The $\sigma $-property in $C(X)$},
url = {http://eudml.org/doc/280140},
volume = {57},
year = {2016},
}

TY - JOUR
AU - Hager, Anthony W.
TI - The $\sigma $-property in $C(X)$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2016
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 57
IS - 2
SP - 231
EP - 239
AB - The $\sigma $-property of a Riesz space (real vector lattice) $B$ is: For each sequence $\lbrace b_{n}\rbrace $ of positive elements of $B$, there is a sequence $\lbrace \lambda _{n}\rbrace $ of positive reals, and $b\in B$, with $\lambda _{n}b_{n}\le b$ 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 $L(X)\le \mathfrak {b}$, then $C(X)$ has $\sigma $. For paracompact $X$, if $C(X)$ has $\sigma $, then $L(X)\le \mathfrak {b}$, and conversely if $X$ is also locally compact. For metrizable $X$, if $C(X)$ has $\sigma $, then $X$is locally compact.
LA - eng
KW - Riesz space; $\sigma $-property; bounding number; $P$-space; paracompact; locally compact
UR - http://eudml.org/doc/280140
ER -