Definition
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
- 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
- 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)