Monotone covering properties defined by closure-preserving operators

We continue Gartside, Moody, and Stares' study of versions of monotone para-compactness. We show that the class of spaces with a monotone open closure-preserving operator is strictly larger than those with a monotone open locally-finite operator. We prove that monotonically metacompact GO-spaces have a monotone open star-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. (C) 2020 Elsevier B.V. All rights reserved.