Monotone orthocompactness and property $(A_o)$
Keywords:
Monotone covering properties, compact, paracompact, orthocompact, monotone orthocompact, property $(A_o)$, GO-spaces, LOTS, $\sigma$-closed discrete dense subset, metrization, stationary setsAbstract
Dedicated to the memory of Phillip L. Zenor, our dear colleague, teacher, and friend, with appreciation for his kindness, good cheer, and willingness to always discuss mathematical ideas.
We continue the study of monotonic orthocompactness with respect to interior preserving open refinements (abbreviated $\mathrm{MO}_o$ ) introduced by Popvassilev and Porter. We show that a GO -space is $\mathrm{MO}_o$ provided that it contains a $\sigma$-closed-discrete set $D$ such that the complement of its closure is $\mathrm{MO}_o$; in particular, Alexandrov's double arrow space as well as the lexicographically ordered square are $\mathrm{MO}_o$. Hence the result of Chase and Gruenhage that compact Hausdorff spaces which are monotonically (countably) metacompact must be metrizable does not extend to $\mathrm{MO}_o$. We show that a compact LOTS which is $\mathrm{MO}_o$ must be first countable, and a monotonically normal space which is $\mathrm{MO}_o$ must be hereditarily paracompact. We also introduce a formally weaker property ( $\mathrm{A}_o$ ) as a useful tool in this study. We show that the one-point compactification of an uncountable discrete space, the Alexandrov duplicate of the unit interval, and stationary subsets of regular uncountable cardinals do not have property ( $\mathrm{A}_o$ ) and hence are not $\mathrm{MO}_o$.
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