Order intervals in $C\left(K\right)$. Compactness, coincidence of topologies, metrizability

Lipecki, Zbigniew. "Order intervals in $C(K)$. Compactness, coincidence of topologies, metrizability." Commentationes Mathematicae Universitatis Carolinae 62 63.3 (2022): 295-306. <http://eudml.org/doc/299039>.

@article{Lipecki2022,
abstract = {Let $K$ be a compact space and let $C(K)$ be the Banach lattice of real-valued continuous functions on $K$. We establish eleven conditions equivalent to the strong compactness of the order interval $[0,x]$ in $C(K)$, including the following ones: (i) $\lbrace x>0\rbrace $ consists of isolated points of $K$; (ii) $[0,x]$ is pointwise compact; (iii) $[0,x]$ is weakly compact; (iv) the strong topology and that of pointwise convergence coincide on $[0,x]$; (v) the strong and weak topologies coincide on $[0,x]$. Moreover, the weak topology and that of pointwise convergence coincide on $[0,x]$ if and only if $\lbrace x>0\rbrace $ is scattered. Finally, the weak topology on $[0,x]$ is metrizable if and only if the topology of pointwise convergence on $[0,x]$ is such if and only if $\lbrace x>0\rbrace $ is countable.},
author = {Lipecki, Zbigniew},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {real linear lattice; order interval; locally solid; Banach lattice $C(K)$; strongly compact; weakly compact; pointwise compact; coincidence of topologies; metrizable; scattered; Čech–Stone compactification},
language = {eng},
number = {3},
pages = {295-306},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Order intervals in $C(K)$. Compactness, coincidence of topologies, metrizability},
url = {http://eudml.org/doc/299039},
volume = {62 63},
year = {2022},
}

TY - JOUR
AU - Lipecki, Zbigniew
TI - Order intervals in $C(K)$. Compactness, coincidence of topologies, metrizability
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2022
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 62 63
IS - 3
SP - 295
EP - 306
AB - Let $K$ be a compact space and let $C(K)$ be the Banach lattice of real-valued continuous functions on $K$. We establish eleven conditions equivalent to the strong compactness of the order interval $[0,x]$ in $C(K)$, including the following ones: (i) $\lbrace x>0\rbrace $ consists of isolated points of $K$; (ii) $[0,x]$ is pointwise compact; (iii) $[0,x]$ is weakly compact; (iv) the strong topology and that of pointwise convergence coincide on $[0,x]$; (v) the strong and weak topologies coincide on $[0,x]$. Moreover, the weak topology and that of pointwise convergence coincide on $[0,x]$ if and only if $\lbrace x>0\rbrace $ is scattered. Finally, the weak topology on $[0,x]$ is metrizable if and only if the topology of pointwise convergence on $[0,x]$ is such if and only if $\lbrace x>0\rbrace $ is countable.
LA - eng
KW - real linear lattice; order interval; locally solid; Banach lattice $C(K)$; strongly compact; weakly compact; pointwise compact; coincidence of topologies; metrizable; scattered; Čech–Stone compactification
UR - http://eudml.org/doc/299039
ER -