On monotone paracompactness

Gartside and Moody proved that a space is protometrizable if and only if it has a monotone star-refinement operator on open covers. They called this property monotone paracompactness but noted that it might be better termed monotone full-normality, and posed the problem of characterizing spaces with a monotone locally-finite operator on open covers. Stares studied related monotone properties but left the above problem open. We introduce Nötherianly locally-finite bases, show that protometrizable spaces have such bases, and that spaces with such bases have a monotone locally-finite operator and are monotonically normal. An example of a non-protometrizable LOTS due to Fuller is shown to have a Nötherianly locally-finite base. The product L ( ω 1 ) × ( ω + 1 ) though not hereditarily normal, has a monotone locally-finite operator, while M × ( ω + 1 ) (where M is the Michael line) does not.