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Cyclic monotonically normal spaces from Cantor sets. (English) Zbl 0878.54011
Mary Ellen Rudin [Proc. Am. Math. Soc. 119, No. 1, 303-307 (1993; Zbl 0789.54027)] constructed a non-acyclic, monotonically normal topological space using a ternary tree. In the paper under review the author first simplifies the Rudin construction to provide a closed subspace \(X\) of the Rudin space such that \(X\) is non-acyclic monotonically normal, separable, clopen-homogeneous, and contains a dense skeleton which is the union of countably many disjoint closed Sorgenfrey lines. Rudin showed that her example is a \(K_0\)-space, but it is not known if \(X\) is a \(K_0\)-space. This question is related to the conjecture that every monotonically normal \(K_0\)-space is acyclic monotonically normal, see [P. J. Moody and A. W. Roscoe, Topology Appl. 47, No. 1, 53-67 (1992; Zbl 0801.54018)]. The remainder of the paper provides a geometric generalization of Rudin’s construction, and considers various properties of the spaces constructed.
MSC:
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
54G20 Counterexamples in general topology
54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
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