*Definition [#j15370ac]
A topological space X is said to be monotonically countably metacompact if there is a function m on the set of countable open covers of X (which is called a monotone metacompactness operator) such that:
+if U is a countable open cover of X, then m(U) is a [[point-finite]] open cover of X which refines U;
+if U and V are countable open covers of X with U refining V, then m(U) refines m(V).
*Remark [#m67c796d]
-See [[monotonically compact]]
*Reference [#t48da965]
-H. R. Bennett, K. P. Hart, and D. J. Lutzer, ''A note on monotonically metacompact spaces'', Topology and its Applications, 157(2010), 456-465.
-http://www.math.wm.edu/~lutzer/drafts/BigBushes.pdf (preprint)