Monotone Covering Properties defined by Closure-Preserving Operators

TL;DR

This paper explores various types of monotone covering properties in topological spaces, establishing new relationships and conditions under which spaces are metrizable, especially focusing on GO-spaces and LOTS.

Contribution

It introduces new distinctions between classes of spaces with monotone closure-preserving and open operators, and links these properties to metrizability in GO-spaces.

Findings

Spaces with a monotone closure-preserving open operator are more general than those with a monotone open locally-finite operator.

Monotonically metacompact GO-spaces possess a monotone open locally-finite operator.

Certain GO-spaces with a $\sigma$-closed-discrete dense subset and a monotone closure-preserving operator are metrizable.

Abstract

We continue Gartside, Moody, and Stares' study of versions of monotone paracompactness. We show that the class of spaces with a monotone closure-preserving open operator is strictly larger than those with a monotone open locally-finite operator. We prove that monotonically metacompact GO-spaces have a monotone open locally-finite operator, and so do GO-spaces with a monotone (open or not) closure-preserving operator, whose underlying LOTS has a $\sigma$-closed-discrete dense subset. A GO-space with a $\sigma$-closed-discrete dense subset and a monotone closure-preserving operator is metrizable. A compact LOTS with a monotone open closure-preserving operator is metrizable.