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Hereditarily paracompact and compact monotonically normal spaces. (English) Zbl 0983.54021

In [Quest. Answers Gen. Topology 4, 117-128 (1987; Zbl 0625.54039)] J. Nikiel has conjectured that every compact monotonically normal space is the continuous image of a compact linearly ordered space. In [Topology Appl. 82, No. 1-3, 397-419 (1998; Zbl 0889.54014); 85, No. 1-3, 319-333 (1998; Zbl 0983.54007), see above] the author has shown that every separable compact monotonically normal space is the continuous image of a compact linearly ordered space. In this paper she strengthens her partial solution of Nikiel’s conjecture by proving that every hereditarily paracompact compact monotonically normal space is the continuous image of a compact linearly ordered space. In the meantime she has settled the problem in its whole generality by proving that Nikiel’s conjecture is true (to appear).

MSC:

54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54B99 Basic constructions in general topology
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References:

[1] Balogh, Z.; Rudin, M. E., Monotone normality, Topology Appl., 47, 115-127 (1992) · Zbl 0769.54022
[2] Gartside, P. M., Cardinal invariants of monotonically normal spaces, Topology Appl., 77, 303-314 (1997) · Zbl 0872.54005
[3] Juhasz, I., Cardinal Functions in Topology, Math. Centre Tracts, 34 (1975), CWI: CWI Amsterdam, p. 115
[4] Nikiel, J., Some problems on continuous images of compact spaces, Questions Answers in Gen. Topology, 4, 117-128 (1986) · Zbl 0625.54039
[5] Rudin, M. E., Compact, separable, linearly ordered spaces, Topology Appl., 82, 397-419 (1998) · Zbl 0889.54014
[6] Rudin, M. E., Zero-dimensionality and monotone normality, Topology Appl., 85, 319-333 (1998) · Zbl 0983.54007
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