Abstract
We revisit monotonically semi-neighborhood refining (MSNR) spaces which were introduced by Stares in 1996. MSNR spaces are shown to be lob-spaces with well-ordered (F). The relationship between MSNR spaces with other monotone covering properties are also explored. We show the existence of MSNR spaces that do not posses a monotone locally-finite refining operator and spaces with a monotone locally-finite refining operator that are not MSNR answering a question of Popvassilev and Porter. Compact MSNR spaces may not be metrizable in general, but compact MSNR LOTS are. GO-spaces whose underlying LOTS has a σ-closed-discrete dense subset are shown to have a monotone star-finite refining operator.
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