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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.

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 |

### Keywords:

continuous image
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\textit{M. E. Rudin}, Topology Appl. 85, No. 1--3, 319--333 (1998; Zbl 0983.54007)

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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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