A Note on Monotonically Metacompact Spaces

TL;DR

This paper explores the properties of monotonically metacompact spaces, especially in generalized ordered spaces, providing characterizations, examples, and answering open questions about metacompactness and metrizability.

Contribution

It establishes that metacompact Moore spaces are monotonically metacompact and characterizes monotone metacompactness in GO-spaces, including new examples and counterexamples.

Findings

Metacompact Moore spaces are monotonically metacompact.

A GO-space with a sigma-closed-discrete dense subset is metrizable iff monotonically (countably) metacompact.

Existence of non-metrizable monotonically metacompact LOTS and examples related to Souslin lines.

Abstract

We show that any metacompact Moore space is monotonically metacompact and use that result to characterize monotone metacompactness in certain generalized ordered (GO)spaces. We show, for example, that a generalized ordered space with a sigma-closed-discrete dense subset is metrizable if and only if it is monotonically (countably) metacompact, that a monotonically (countably) metacompact GO-space is hereditarily paracompact, and that a locally countably compact GO-space is metrizable if and only if it is monotonically (countably) metacompact. We give an example of a non-metrizable LOTS that is monotonically metacompact, thereby answering a question posed by S. G. Popvassilev. We also give consistent examples showing that if there is a Souslin line, then there is one Souslin line that is monotonically countable metacompact, and another Souslin line that is not monotonically countably metacompact.