Monotonically compact and monotonically Lindelöf spaces
by
Gary Gruenhage
Auburn University

A space X is monotonically compact (resp., monotonically Lindelöf) if for each open cover U, one can assign a finite (resp., countable) open refinement r(U) of U such that r(V) refines r(U) whenever V refines U. It is easily seen that compact metrizable spaces are monotonically compact and separable metrizable spaces are monotonically Lindelöf.

Bennett and Lutzer have asked whether every monotonically compact linearly ordered space is metrizable. We show that the answer is "yes". But it seems to be unknown if there are any nonmetrizable monotonically compact Hausdorff spaces at all. We show that there aren't any scattered ones.

We also show that any first countable Lindelöf generalized ordered space is monotonically Lindelöf. This answers a couple of questions of Bennett, Lutzer, and Matveev concerning monotone Lindelöfness of Suslin lines.

Date received: February 21, 2008