Monotone Covering Properties defined by Closure-Preserving Operators

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.