×

Nikiel’s conjecture. (English) Zbl 0988.54022

Monotonically normal spaces exhibit a lot of structure, so much so that J. Nikiel [Quest. Answers Gen. Topology 4, 117-128 (1987; Zbl 0625.54039)] conjectured that every compact monotonically normal space is the continuous image of a compact ordered space. The present paper offers a proof of this conjecture, by induction on the density of the space and with a new proof for the separable case (which was dealt with in the author’s paper [Topology Appl. 82, No. 1-3, 397-419 (1998; Zbl 0889.54014)]). Monotone normality can be defined by means of an operator \(H\) that assigns to every pair \((x,U)\) with \(x\in U\) and \(U\) open an open set \(H(x,U)\) with (i) if \(x\notin V\) and \(y\notin U\) then \(H(x,U)\cap H(y,V)=\emptyset\), and (ii) \(x\in H(x,U)\subseteq U\). Using this operator the author constructs what she calls break-downs, which are families of closed sets with a tree-like flavour. The ordered space is then obtained as a kind of branch space of this structure. This outline belies the intricate nature of the proof.
Reviewer: K.P.Hart (Delft)

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
54A35 Consistency and independence results in general topology
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Balogh, Z.; Rubin, 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] Heath, R. W.; Lutzer, D. L.; Zenor, P. L., Monotonically normal spaces, Trans. Amer. Math. Soc., 178, 481-493 (1973) · Zbl 0269.54009
[4] Juhasz, I., Cardinal functions in topology, (Math. Centre Tracts, 34 (1983), CWI: CWI Amsterdam), 114-118, Section (A.4)
[5] P.J. Moody, Acyclic monotonically normal spaces, Thesis, Mathematical Institute, Oxford, 1988; P.J. Moody, Acyclic monotonically normal spaces, Thesis, Mathematical Institute, Oxford, 1988
[6] Nikiel, J., Some problems on continuous images of compact ordered spaces, Question Answers Gen. Topology, 4, 117-128 (1986) · Zbl 0625.54039
[7] Nikiel, J.; Purisch, S.; Treybig, L. B., Separable zero-dimensional spaces which are continuous images or ordered compacta, Houston J. Math., 24, 45-56 (1998) · Zbl 0965.54034
[8] Nikiel, J.; Treybig, L. B.; Tuncali, H. M., Local connectivity and maps onto non-metrizable arcs, Internat. J. Math. Math. Sci., 20, 4, 681-688 (1997) · Zbl 0889.54021
[9] Rudin, M. E., Compact separable linearly ordered spaces, Topology Appl., 82, 397-419 (1998) · Zbl 0889.54014
[10] Rudin, M. E., Zero dimensionality and monotone normality, Topology Appl., 85, 319-333 (1998) · Zbl 0983.54007
[11] Williams, S. W.; Zhou, H., Strong versions of normality, (Topology and Its Applications. Topology and Its Applications, Lectures Pure Appl. Math., 135 (1991), Marcel Dekker: Marcel Dekker New York) · Zbl 0797.54011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.