Abstract
In this paper, we show that any generalized ordered space X is monotonically (countably) metacompact if and only if the subspace X - { x } is monotonically (countably) metacompact for every point x of X and monotone (countable) metacompact property is hereditary with respect to convex (open) subsets in generalized ordered spaces. In addition, we show the equivalence of two questions posed by H.R. Bennett, K.P. Hart and D.J. Lutzer.
References
[1] Bennett H.R., Hart K.P., Lutzer D.J., A note on monotonically metacompact spaces, Topology Appl., 2010, 157, 456-465 10.1016/j.topol.2009.10.004Search in Google Scholar
[2] Engleking R., General Topology, 2nd ed., Sigma Ser. Pure Math., 6, Hedermann, Berlin, 1989 Search in Google Scholar
[3] Kemoto N., Normality of Products of GO-spaces and cardinals, Topology Proc., 1993, 18, 133-142 Search in Google Scholar
[4] Lutzer D.J., Ordered topological spaces, In: Surveys in General Topology, Reed G.M., Academic Press, New York, 1980, 247-296 10.1016/B978-0-12-584960-9.50014-6Search in Google Scholar
[5] Popvassilev S.G., !1 C1 is not monotonically countably compact, Questions Answers Gen. Topology, 2009, 27, 133-135 Search in Google Scholar
[6] Xu A.J., Shi W.X., Notes on monotone Lindelöf property, Czechoslovak Math. J., 2009, 59, 943-955 10.1007/s10587-009-0065-8Search in Google Scholar
[7] Xu A.J., Shi W.X., Monotone Lindelöf property and linearly ordered extensions, Bull. Aust. Math. Soc., 2010, 81, 418-424 10.1017/S0004972709001026Search in Google Scholar
[8] Xu A.J., Shi W.X., A result on monotonically metacompact spaces, Houston J. Math., 2013, 39, 1437-1442 Search in Google Scholar
©2015 Ai-Jun Xu
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.
