Some minimum topological spaces, and vector lattices

TL;DR

This paper explores the existence of minimal topological spaces and vector lattices under certain covering and hull operators, linking topological properties with algebraic structures via the Yosida representation theorem.

Contribution

It establishes conditions for the existence of minimal compact Hausdorff spaces and vector lattices related to specific covering and hull operators, extending the understanding of their interrelations.

Findings

Existence results for minimal compact Hausdorff spaces under various operators.

Connections between topological minimality and algebraic vector lattice structures.

Specific cases analyzed include Gleason, quasi-F, uniform, and essential completion operators.

Abstract

We investigate the existence of compact Hausdorff spaces $X$ that are minimum with respect to $cX=K$ for some fixed covering operator $c$ and compact Hausdorff space $K$ with $cK=K$. Then, using the Yosida representation theorem, we show how that situation relates to the existence of Archimedean vector lattices $A$ with distinguished strong unit that are minimum with respect to $hA=H$ for some fixed hull operator $h$ and vector lattice $H$ with $hH=H$. Among others, we obtain answers for $c=g$ (the Gleason covering operator), $c=qF$ (the quasi-$F$ covering operator), $h = u$ (the uniform completion operator), and $h=e$ (the essential completion operator).