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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)
##### Keywords:
monotonically normal topological space; non-acyclic
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##### References:
 [1] Arhangel’skii, A.V., Mappings and spaces, Russian math. surveys, 21, 115-162, (1966) · Zbl 0171.43603 [2] Burke, D.K.; van Douwen, E.K., No dense metrizable Gδ-subspaces in butterfly semimetrizable Baire spaces, Topology appl., 11, 31-36, (1980) · Zbl 0415.54016 [3] Comfort, W.W.; Hofmann, K.H.; Remus, D., Topological groups and semigroups, (), 57-144 · Zbl 0798.22001 [4] Comfort, W.W.; Kato, A.; Shelah, S., Topological partition relation of the form ω* → (Y)21, (), 70-79 · Zbl 0855.54004 [5] K. Eda, Personal communication. [6] Gruenhage, G., Generalized metric spaces, (), 423-501 [7] Gruenhage, G., Perfectly normal compacta, cosmic spaces and some partition problems, (), 85-95 [8] Gruenhage, G., Generalized metric spaces and metrization, (), 239-274 · Zbl 0794.54034 [9] Heath, R.W.; Lutzer, D.; Zenor, P., Monotonically normal spaces, Trans. amer. math. soc., 178, 481-493, (1973) · Zbl 0269.54009 [10] Kato, A., A new construction of extremally disconnected topologies, Topology appl., 58, 1-16, (1994) · Zbl 0804.54030 [11] Moody, P.J.; Roscoe, A.W., Acyclic monotone normality, Topology appl., 47, 53-67, (1992) · Zbl 0801.54018 [12] Motorov, D.B., Homogeneity and π-networks, Vestnik moskov. univ. ser. I mat. mekh., 44, 31-34, (1989) · Zbl 0708.54017 [13] Ostaszewski, A.J., Monotone normality and Gδ-diagonals in the class of inductively generated spaces, (), 905-930 · Zbl 0459.54021 [14] Rudin, M.E., A cyclic monotonically normal space which is not K_{0}, (), 303-307 · Zbl 0789.54027 [15] Ruppert, W.A.F., Compact semitopological semigroups, (), 133-170 · Zbl 0713.22003 [16] Steen, L.A.; Seebach, J.A., Counterexamples in topology, (1978), Springer · Zbl 0211.54401 [17] Terada, T., Spaces whose all nonempty clopen subspaces are homeomorphic, Yokohama math. J., 40, 87-93, (1993) · Zbl 0819.54006 [18] Watson, S.; Weiss, W., A topology on the union of the double arrow space and the integers, Topology appl., 28, 177-179, (1988) · Zbl 0632.54023 [19] Watson, S., The construction of topological spaces, (), 673-757 · Zbl 0803.54001
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