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On Homogeneous Images of Compact Ordered Spaces

Published online by Cambridge University Press:  20 November 2018

J. Nikiel  and
E.D. Tymchatyn
J. Nikiel
Affiliation:
Department of Mathematics, University of Saskatchewan, Saskatoon Saskatchewan, S7N 0W0
E.D. Tymchatyn
Affiliation:
Department of Mathematics, University of Saskatchewan, Saskatoon Saskatchewan, S7N 0W0
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Abstract

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We answer a 1975 question of G R Gordh by showing that if X is a homogeneous compactum which is the continuous image of a compact ordered space then at least one of the following holds

(I) X is metrizable, (II) dim X = 0 or (III) X is a union of finitely many pairwise disjoint generalized simple closed curves.

We begin to examine the structure of homogeneous 0-dimensional spaces which are continuous images of ordered compacta.

Keywords

54F0554C05arccompact ordered spacecontinuous imagehomogeneousstrongly homogeneousdendron
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 2 , 01 April 1993 , pp. 380 - 393
Copyright
Copyright © Canadian Mathematical Society 1993

References

1. Babcock, W., On linearly ordered topological spaces, Ph. D. dissertation, Tulane University, 1964.Google Scholar
2. Bell, M.G., Non-homogeneity of powers of COR images, preprint, 1989.Google Scholar
3. Charatonik, J.J., Open mappings of universal dendrites, Bull. Acad. Polon. Sci., ser. sci. math. 28(1980), 489–494.Google Scholar
4. Devlin, K.J. and H. Johnsbraten, The Souslin problem, Lecture Notes in Mathematics 405, Springer Verlag, 1974.Google Scholar
5. Engelking, R., General topology, Polish Scientific Publishers, Warsaw, 1977.Google Scholar
6. Fedorcuk, V.V., Strongly closed mappings, Soviet Math. Dokl. 10(1969), 804–806.Google Scholar
7. Gordh, G.R., Jr., On homogeneous hereditarily unicoherent continua, Proc. Amer. Math. Soc. 51(1975), 198–202.Google Scholar
8. Hart, K.P. and van Mill, J., A method for constructing ordered continua, Topology Appl. 21(1985), 35–49.Google Scholar
9. Kuratowski, K., Topology, vol. II, Academic Press, New York, 1968.Google Scholar
10. Mardešić, S., Images of ordered compacta are locally peripherally metric, Pacific J. Math. 23(1967), 557–568.Google Scholar
11. Maurice, M.A., Compact ordered spaces, Math. Centre Tracts 6, Amsterdam, 1964.Google Scholar
12. van Mill, J., Characterization of some zero-dimensional separable metric spaces, Trans. Amer. Math. Soc. 264(1981), 205–215.Google Scholar
13. van Mill, J. and Wattel, E., Dendrons, Topology and order structures, I, Math. Centre Tracts 142, Amsterdam, 1981,59-81.Google Scholar
14. Mohler, L. and Oversteegen, L.G., On hereditarily decomposable hereditarily equivalent non-metric continua, Fund. Math. 136(1990), 1–12.Google Scholar
15. Nikiel, J., Some problems on continuous images of compact ordered spaces, Questions Answers Gen. Topology 4(1986/87), 117–128.Google Scholar
16. Nikiel, J., Images of arcs—a nonseparable version of the Hahn-Mazurkiewicz theorem, Fund. Math. 129(1988),91-120.Google Scholar
1. Nikiel, J., A continuous partial ordering for images of arcs. In: General Topology and its Relations to Modern Analysis and Algebra, VI, Proc. Sixth Prague Topological Symp. 1986, (ed. Frolik, Z.), Heldermann Verlag, Berlin, 1988,361-370.Google Scholar
18. Nikiel, J., Orderability properties of a zero-dimensional space which is a continuous image of an ordered compactum, Topology Appl. 31(1989), 269–276.Google Scholar
19. Nikiel, J. , Topologies on pseudo-trees and applications, Mem. Amer. Math. Soc. (416) 82(1989), 1–116.Google Scholar
20. Nikiel, J. , On continuous images of arcs and compact orderable spaces, Topology Proc. 14(1989), 163–193.Google Scholar
21. Purisch, S., Williams, S.W. and Haoxuan Zhou, Continuous images of compact orderable spaces and monotonically normal spaces, preprint, 1990.Google Scholar
22. Treybig, L.B., Concerning homogeneity in totally ordered, connected topological space, Pacific J. Math. 13(1963), 1417–1421.Google Scholar
23. Treybig, L.B., Separation by finite sets in connected, continuous images of ordered compacta, Proc. Amer. Math. Soc. 74(1979), 326–328.Google Scholar
24. Treybig, L.B., Arcwise connectivity in continuous images of ordered compacta, Glasnik Mat. 21(1986), 201 -211.Google Scholar
25. Wazewski, T., Sur les courbes de Jordan ne refermant ancune courbe fermée de Jordan, Ann. Soc. Pol. Math. 2(1923), 49–170.Google Scholar
26. Whyburn, G.T., Analytic topology, Amer. Math. Soc, Providence, 1942.Google Scholar
27. Whyburn, G.T., Cut points in general topological spaces, Proc. Nat. Acad. Sci. USA 61(1968), 380–387.Google Scholar
28. van Mill, J. and Wattel, E., Subbase characterizations of subspaces of compact trees, Topology Appl. 13(1982), 321–326.Google Scholar