Abstract
LetX be an Hausdorff space. We say thatX is a CO space, ifX is compact and every closed subspace ofX is homeomorphic to a clopen subspace ofX, andX is a hereditarily CO space (HCO space), if every closed subspace is a CO space. It is well-known that every well-ordered chain with a last element, endowed with the interval topology, is an HCO space, and every HCO space is scattered. In this paper, we show the following theorems:
Theorem (R. Bonnet):
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(a)
Every HCO space which is a continuous image of a compact totally disconnected interval space is homeomorphic to β+1 for some ordinal β.
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(b)
Every HCO space of countable Cantor-Bendixson rank is homeomorphic to α+1 for some countable ordinal α.
Theorem (S. Shelah):Assume \(\diamondsuit _{\aleph _1 } \). Then there is a HCO compact space X of Cantor-Bendixson rankω 1} and of cardinality ℵ1 such that:
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(1)
X has only countably many isolated points,
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(2)
Every closed subset of X is countable or co-countable,
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(3)
Every countable closed subspace of X is homeomorphic to a clopen subspace, and every uncountable closed subspace of X is homeomorphic to X, and
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(4)
X is retractive.
In particularX is a thin-tall compact space of countable spread, and is not a continuous image of a compact totally disconnected interval space. The question whether it is consistent with ZFC, that every HCO space is homeomorphic to an ordinal, is open.
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References
U. Avraham and R. Bonnet,Every superatomic subalgebra of an interval algebra is embeddable in an ordinal algebra, Proc. Amer. Math. Soc.115 (1992), 585–592.
Z. Balogh, A. Dow, D. H. Fremlin and P. J. Nyikos,Countable tightness and proper forcing, Bull. Amer. Math. Soc.19, (1988), 295–298.
M. Bekkali,On superatomic Boolean algebras, Ph.D. Dissertation, University of Colorado, Boulder, viii+122pp.
M. Bekkali, R. Bonnet and M. Rubin,Compact interval spaces in which all closed subset are homeomorphic to clopen ones [I], Order9 (1993), 69–95.
M. Bekkali, R. Bonnet and M. Rubin,Compact interval spaces in which all closed subset are homeomorphic to clopen ones [II], Order9 (1993), 177–200.
R. Bonnet,On superatomic Boolean algebras to appear inProc. of Finite and Infinite Combinatorics and Logic, 1991, Banff (Canada), Ed. N. Sauer, NATO ASI Publ.
R. Bonnet and M. Rubin,On Boolean algebras which are isomorphic to each of it uncountable subalgebra, submitted to Canad. J. Math.
R. Bonnet, M. Rubin and H. Si-Kaddour,On Boolean algebras with well-founded sets of generators, 2nd version, submitted to Trans. Amer. Math. Soc.
G. W. Day,Superatomic Boolean algebras, Pacific J. Math.23 (1967), 479–489.
L. Henkin, J.D. Monk and A. Tarski,Cylindric Algebras, Part I, North-Holland, Amsterdam, New York, 1971.
T. Jech,Set Theory, Pure and Applied Mathematics, Academic Press, New York, 1978.
S. Koppelberg,Special classes of Boolean algebras inHandbook on Boolean Algebras, Vol. 1, Part I, chapter 6, 239–284 (Ed. J. D. Monk, with the collaboration of R. Bonnet), North-Holland, Amsterdam, 1989, pp. 239–284.
K. Kunen,Set Theory, Studies in Math. Logic, North-Holland, 1980.
S. Mazurkiewicz and W. Sierpinski,Contribution à la topologie des ensembles dénombrables, Fund. Math.1 (1920), 17–27.
J. D. Monk,Appendix in Set Theory, inHandbook on Boolean Algebras, Vol. 3, Part II, Section D (Ed. J. D. Monk, with the collaboration of R. Bonnet), North Holland, Amsterdam, 1989, pp. 1213–1233.
A. Ostaszewski,A perfectly normal countably compact scattered space which is strongly zero-dimensional. J. London Math. Soc.14 (1976), 167–177.
A. Ostaszewski,On countably compact, perfectly normal spaces, J. London Math. Soc.14 (1976), 505–516.
J. Roitman,Superatomic Boolean algebras, inHandbook on Boolean Algebras, Vol. 3, Part II, chapter 19 (Ed. J. D. Monk, with the collaboration of R. Bonnet), North-Holland, Amsterdam, 1989, pp. 719–740.
J. Roitman,A space homeomorphic to each of its uncountable closed subspace under CH, preprint.
J. G. Rosenstein,Linear Ordering, Academic Press, New York, 1982.
M. E. Rudin,Lectures on set theoretic topology, Conference Series in Math., No. 23, Amer. Math. Soc., Providence, RI, 1975.
J. Steprans,Steprans's problems, inOpen Problems in Topology (J. Van Mill and G.M. Reed, eds.), North-Holland, Amsterdam, 1990, pp. 13–20.
M. Weese,Subalgebra-rigid and homomorphism-rigid Boolean algebras, manuscript.
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Supported by the Ministère Français des Affaires Etrangères, and the Ben Gurion University of the Negev. CNRS, URA 753: Equipe de Logique Mathématique (Paris VII).
Supported by the United States-Israel Binational Science Foundation. Publication No. 359.
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Bonnet, R., Shelah, S. On HCO spaces. An uncountable compactT 2 space, different from ℵ1+1, which is homeomorphic to each of its uncountable closed subspaces, which is homeomorphic to each of its uncountable closed subspaces. Israel J. Math. 84, 289–332 (1993). https://doi.org/10.1007/BF02760945
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DOI: https://doi.org/10.1007/BF02760945