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Monotone covering properties defined by closure-preserving operators. (English) Zbl 1470.54013

A space \(X\) is said to have a monotone (respectively, monotone open) closure-preserving operator \(r\) if whenever \(\mathcal U\) is an open cover of \(X\), then \(r(\mathcal U)\) is a closure-preserving cover of arbitrary (respectively, open) sets that refines \(\mathcal U\) and such that if \(\mathcal U\) and \(\mathcal V\) are open covers of \(X\) such that \(\mathcal U\) refines \(\mathcal V\), then \(r(\mathcal U)\) refines \(r(\mathcal V)\). After a short introduction, Section 2 of this paper deals with monotone operators in \(GO\)-spaces. For a \(GO\)-space \(X\), \(E_r\) denotes the set \(\{x\in X:\) neither \([x,\rightarrow)\) nor \((\leftarrow, x]\) is open\(\}\), and then one of the main results of this section, Theorem 2.7, states that if \(X\) is a \(GO\)-space with a monotone closure-preserving operator, and if \(E_r\) is closed and discrete, then \(X\) has a monotone open closure-preserving operator. Another result of this section, Theorem 2.10, includes among many other statements, that if \((X,\tau,<)\) is a \(GO\)-space such that the order topology of \((X,<)\) has a strongly \(\sigma\)-discrete dense subspace, then \(X\) has a monotone closure-preserving operator if and only if it has a monotone open closure-preserving operator. Section 3 deals with the metrization of \(LOTS\) and \(GO\)-spaces. Among the principal results of this section, it is shown that if \(X\) is a compact \(LOTS\) with a monotone closure-preserving operator, then \(X\) is first countable and if \(X\) has a monotone open closure-preserving operator then \(X\) is metrizable. An interesting unsolved problem is whether or not the word “open” can be omitted in the last result.

MSC:

54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
54E35 Metric spaces, metrizability
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[1] Bennett, H.; Hart, K.; Lutzer, D., A note on monotonically metacompact spaces, Topol. Appl., 157, 2, 456-465 (2010) · Zbl 1184.54023
[2] Bennett, H.; Lutzer, D.; Purisch, S., On dense subspaces of generalized ordered spaces, Topol. Appl., 93, 3, 191-205 (1999) · Zbl 0942.54026
[3] Chase, T.; Gruenhage, G., Compact monotonically metacompact spaces are metrizable, Topol. Appl., 160, 1, 45-49 (2013) · Zbl 1264.54039
[4] Chase, T.; Gruenhage, G., Monotone covering properties and properties they imply, Topol. Appl., 213, 135-144 (2016) · Zbl 1356.54031
[5] Engelking, R., General Topology (1989), Heldermann Verlag: Heldermann Verlag Berlin · Zbl 0684.54001
[6] Faber, M. J., Metrizability in Generalized Ordered Spaces, Mathematical Centre Tracts, vol. 53 (1974), Mathematisch Centrum: Mathematisch Centrum Amsterdam · Zbl 0282.54017
[7] Gartside, P. M.; Moody, P. J., A note on protometrizable spaces, Topol. Appl., 52, 1, 1-9 (1993) · Zbl 0797.54030
[8] Gruenhage, G., Monotonically compact and monotonically Lindelöf spaces, Quest. Answ. Gen. Topol., 26, 121-130 (2008) · Zbl 1163.54017
[9] Gruenhage, G., Monotonically compact Hausdorff spaces are metrizable, Quest. Answ. Gen. Topol., 27, 1, 57-59 (2009) · Zbl 1173.54011
[10] Lutzer, D., Ordered topological spaces, (Reed, G. M., Surveys in General Topology (1980), Acad. Press: Acad. Press New York), 247-295 · Zbl 0472.54020
[11] Nyikos, P., Some surprisings base properties in topology, Part I, (Stavrakas; Allen, Studies in Topology (1975), Acad. Press), 427-450 · Zbl 0337.54014
[12] Peng, L.; Li, H., A note on monotone covering properties, Topol. Appl., 158, 13, 1673-1678 (2011) · Zbl 1239.54012
[13] Popvassilev, S., \( \omega_1 + 1\) is not monotonically countably metacompact, Quest. Answ. Gen. Topol., 27, 2, 133-135 (2009) · Zbl 1182.54028
[14] Popvassilev, S., Versions of monotone covering properties, (29th Summer Conference on Topology and Its Applications (July 2014)), College of Staten Island, CUNY. Abstract retrieved from
[15] Popvassilev, S.; Porter, J., On monotone paracompactness, Topol. Appl., 167, 1-9 (2014) · Zbl 1297.54048
[16] Stares, I., Versions of monotone paracompactness, (Papers on General Topology and Applications. Papers on General Topology and Applications, Gorham, ME, 1995. Papers on General Topology and Applications. Papers on General Topology and Applications, Gorham, ME, 1995, Ann. New York Acad. Sci., vol. 806 (1996), New York Acad. Sci.: New York Acad. Sci. New York), 433-437 · Zbl 0896.54009
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