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Zero dimensionality and monotone normality. (English) Zbl 0983.54007

From the introduction: J. Nikiel [Quest. Answers Gen. Topology 4, 117-128 (1987; Zbl 0625.54039)] has conjectured that every compact monotonically normal space \(X\) is the continuous image of a compact linearly ordered space. In [Topology Appl. 82, No. 1-3, 397-419 (1998; Zbl 0889.54014)] the author proved this conjecture if \(X\) is also separable and zero-dimensional. In this paper she proves the conjecture for separable \(X\) by proving the
Theorem. Suppose \(X\) is a separable, compact, monotonically normal space. Then there is a separable, compact, zero-dimensional, monotonically normal space \(\Delta\) and a continuous map \(\pi\) from \(\Delta\) onto \(X\).
The rest of the paper consists of a proof of this theorem. It mimics the proof given in [the author, loc. cit.] without assuming zero-dimensionality. The author’s hope is that generalizing the proof in this way may help someone to prove Nikiel’s conjecture.

MSC:

54A35 Consistency and independence results in general topology
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
54F65 Topological characterizations of particular spaces
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References:

[1] Nikiel, J., Some problems on continuous images of compact ordered spaces, Questions Answers Gen. Topology, 4, 117-128 (1986) · Zbl 0625.54039
[2] Rudin, M. E., Compact, separable, linearly ordered spaces, Topology Appl., 82, 397-419 (1998) · Zbl 0889.54014
[3] Heath, R. W.; Lutzer, D. J.; Zener, P. L., Monotonically normal spaces, Trans. Amer. Math. Soc., 178, 481-493 (1973) · Zbl 0269.54009
[4] Ramsey, F. P., On a problem of formal logic, (Proc. London Math. Soc., 30 (1930)), 264-286 · JFM 55.0032.04
[5] Ostaszewski, A. J., Monotone normality and \(G_δ\)-diagonals in the class of inductively generated spaces, (Császár, A., Topology. Topology, Colloquia Mathematica Societatis János Bolyai, 23 (1980), North-Holland: North-Holland Amsterdam), 905-930 · Zbl 0459.54021
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