On the metrization of PIGO spaces

Properties weaker than monotone countable metacompactness are studied in PIGO and compact spaces. GO-spaces with a σ-closed-discrete dense subset and the NZ(ω)-property are metrizable generalizing results of Bennett, Hart and Lutzer and Peng and Li on monotonically countably metacompact spaces. NSR pair-families are introduced, and pair-bases that are NSR pair-families or countable unions of such pair-families are studied. By modifying results of Chase and Gruenhage, we show if a space X with a σ-NSR pair-base is compact, then X is metrizable, generalizing recent results by Chase and Gruenhage and some older results of Gruenhage and Nyikos. We show PIGO spaces with a σ-closed-discrete dense subset and a σ-NSR pair-base are metrizable. Relationships between these properties and others in the literature are also established.