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**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.) |

Full Text:
DOI

### References:

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