Chain intersection closures

TL;DR

This paper investigates the conditions under which spherical completeness is preserved in ball spaces, introducing ultra-diameters and analyzing their impact on ultrametric spaces with partially ordered value sets.

Contribution

It introduces the concept of ultra-diameters and examines their role in preserving spherical completeness under expansions in ultrametric spaces.

Findings

Ultra-diameters can help preserve spherical completeness in certain cases.

Chain intersection closures may fail to preserve spherical completeness in general.

Positive results are obtained for narrow partially ordered sets.

Abstract

We study spherical completeness of ball spaces and its stability under expansions. We introduce the notion of an ultra-diameter, mimicking diameters in ultrametric spaces. We prove some positive results on preservation of spherical completeness involving ultra-diameters with values in narrow partially ordered sets. Finally, we show that in general, chain intersection closures of ultrametric spaces with partially ordered value sets do not preserve spherical completeness.