A note on monotonically metacompact spaces

Abstract

MSC: primary 54D20 secondary 54E30, 54E35, 54F05 Keywords: Metacompact Countably metacompact Monotonically metacompact Monotonically countably metacompact Generalized ordered space GO-space LOTS Metacompact Moore space Metrizable σ -Closed-discrete dense setWe show that any metacompact Moore space is monotonically metacompact and use that result to characterize monotone metacompactness in certain generalized ordered (GO) spaces. We show, for example, that a generalized ordered space with a σ -closeddiscrete dense subset is metrizable if and only if it is monotonically (countably) metacompact, that a monotonically (countably) metacompact GO-space is hereditarily paracompact, and that a locally countably compact GO-space is metrizable if and only if it is monotonically (countably) metacompact. We give an example of a non-metrizable LOTS that is monotonically metacompact, thereby answering a question posed by S.G. Popvassilev. We also give consistent examples showing that if there is a Souslin line, then there is one Souslin line that is monotonically countable metacompact, and another Souslin line that is not monotonically countably metacompact.

Notes on monotonically metacompact generalized ordered spaces

“…; x/ of X . If H is a stationary subset, it is a contradiction since X is hereditarily paracompact by [1,Proposition 3.4]. Therefore H is not stationary in OE0; 0-cf.x//.…”

“…The operator r is called a monotone (countable) metacompactness operator for X . The concepts were first introduced in [1,5]. H.R.…”

“…Hart and D.J. Lutzer proved that any metacompact Moore space is monotonically metacompact, any monotonically (countably) metacompact generalized ordered space is hereditarily paracompact [1]. They also posed some questions on monotone (countable) metacompactness: Question 4.7]).…”

A note on monotone covering properties

“…In [10,3], the concepts of monotonically countably metacompact and monotonically metacompact were introduced. A space X is monotonically (countably) metacompact if there is a function r that associates with each (countable) open cover U of X an open point-finite refinement r(U ) that covers X , where r has the property that if U and V are open covers with U ≺ V then r(U ) ≺ r(V) (cf.…”

“…A space X is monotonically (countably) metacompact if there is a function r that associates with each (countable) open cover U of X an open point-finite refinement r(U ) that covers X , where r has the property that if U and V are open covers with U ≺ V then r(U ) ≺ r(V) (cf. [10,3]). In [3], it was proved that any metacompact Moore space is monotonically metacompact.…”