Monotonic Covering Properties

Abstract

Monotonic versions of classical topological properties have been of some interest for several years. That adding monotonic to a covering property makes it stronger is well-known, but exactly how much stronger is still, in many cases, unknown. Here we trace the progression of results regarding monotonic covering properties, and then look at monotonic versions of metacompactness and meta-Lindel\"{o}fness, providing a useful property these spaces exhibit, and from that obtain several original results, many of which answer open questions. We strengthen a theorem of G. Gruenhage, (that monotonically compact, $T_2$ spaces are metrizable) by showing that compact, $T_2$, monotonically metacompact spaces are metrizable. We also show that every monotonically metacompact space is hereditarily metacompact, and show by way of a counterexample that Bennett, Hart, and Lutzer's theorem that every regular, developable, metacompact space is monotonically metacompact cannot have the condition ``developable'' weakened to ``quasi-developable'', or replaced by ``stratifiable''. In examining several different spaces for their relationship to this property, we also show that a monotonically Lindel\"{o}f space need not be monotonically metacompact.