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Compactness of order intervals in a locally solid linear lattice

Volume 168 / 2022

Zbigniew Lipecki Colloquium Mathematicum 168 (2022), 297-309 MSC: 46A40, 46B42, 46E05, 54C35, 54D30, 06E99. DOI: 10.4064/cm8624-11-2021 Published online: 9 January 2022


Let $X$ be a linear lattice, let $x\in{X_+} $, and let $\tau $ be a locally solid topology on $X$. We present four conditions equivalent to the {$\tau $-compactness of the order interval $[0,x]$ in $X$, including the following ones: (i) there is a set $S$ and an affine homeomorphism of $[0,x]$ onto the Tychonoff cube $[0,1]^S$ which preserves order; (ii) ${\cal C}_x $, the set of components of $x$, is $\tau $-compact and $[0,x]$ is order $\sigma $-complete. In the special case where $X$ is a Banach lattice and $\tau $ is its norm topology, another equivalent condition is: (iii) ${\cal C}_x $ is weakly compact and $[0,x]$ is order $\sigma $-complete.


  • Zbigniew LipeckiInstitute of Mathematics
    Polish Academy of Sciences
    Wrocław Branch
    Kopernika 18
    51-617 Wrocław, Poland

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