*Definition [#t1f76e72]
A topological space X is said to be monotonically compact if there is a function m on the set of open covers of X (which is called a monotone compactness operator) such that:
+if U is an open cover of X, then m(U) is a finite open cover of X which refines U;
+if U and V are open covers of X with U refining V, then m(U) refines m(V).
*Remark [#o2e050fc]
-See [[monotonically paracompact]], [[monotonically metacompact]], [[monotonically orthocompact]] and [[monotonically ultraparacompact]].
-If the finiteness condition of m(U) in the above definition is changed to the requirement that m(U) must be countable, then one has the definition of monotonically Lindeloef.
*Reference [#j3940fff]
-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)