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A compact group which is not Valdivia compact

Authors: Wieslaw Kubis and Vladimir Uspenskij
Journal: Proc. Amer. Math. Soc. 133 (2005), 2483-2487
MSC (2000): Primary 54D30; Secondary 54C15, 22C05
Published electronically: February 25, 2005
MathSciNet review: 2138892
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Abstract: A compact space $K$ is Valdivia compact if it can be embedded in a Tikhonov cube $I^A$ in such a way that the intersection $K\cap\Sigma$ is dense in $K$, where $\Sigma$ is the sigma-product ($=$ the set of points with countably many non-zero coordinates). We show that there exists a compact connected Abelian group of weight $\omega_1$ which is not Valdivia compact, and deduce that Valdivia compact spaces are not preserved by open maps.

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  • 1. S. Argyros, S. Mercourakis, and S. Negrepontis, Functional-analytic properties of Corson-compact spaces, Studia Math. 89 (1988), no. 3, 197–229. MR 956239
  • 2. M. BURKE, W. KUBIS, S. TODORSCEVIC, Kadec norms on spaces of continuous functions, preprint.
  • 3. Robert Deville and Gilles Godefroy, Some applications of projective resolutions of identity, Proc. London Math. Soc. (3) 67 (1993), no. 1, 183–199. MR 1218125, 10.1112/plms/s3-67.1.183
  • 4. Ryszard Engelking, Theory of dimensions finite and infinite, Sigma Series in Pure Mathematics, vol. 10, Heldermann Verlag, Lemgo, 1995. MR 1363947
  • 5. László Fuchs, Infinite abelian groups. Vol. I, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. MR 0255673
  • 6. László Fuchs, Infinite abelian groups. Vol. II, Academic Press, New York-London, 1973. Pure and Applied Mathematics. Vol. 36-II. MR 0349869
  • 7. Roger Godement, Topologie algébrique et théorie des faisceaux, Hermann, Paris, 1973 (French). Troisième édition revue et corrigée; Publications de l’Institut de Mathématique de l’Université de Strasbourg, XIII; Actualités Scientifiques et Industrielles, No. 1252. MR 0345092
  • 8. Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations, Die Grundlehren der mathematischen Wissenschaften, Bd. 115, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 0156915
  • 9. Ondřej Kalenda, Embedding of the ordinal segment [0,𝜔₁] into continuous images of Valdivia compacta, Comment. Math. Univ. Carolin. 40 (1999), no. 4, 777–783. MR 1756552
  • 10. Ondřej Kalenda, A characterization of Valdivia compact spaces, Collect. Math. 51 (2000), no. 1, 59–81. MR 1757850
  • 11. Ondřej F. K. Kalenda, Valdivia compact spaces in topology and Banach space theory, Extracta Math. 15 (2000), no. 1, 1–85. MR 1792980
  • 12. W. KUBIS, H. MICHALEWSKI, Small Valdivia compact spaces, preprint.
  • 13. Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
  • 14. Z. Frolík (ed.), General topology and its relations to modern analysis and algebra. VI, Research and Exposition in Mathematics, vol. 16, Heldermann Verlag, Berlin, 1988. MR 952585
  • 15. V. V. Uspenskiĭ, Topological groups and Dugundji compact spaces, Mat. Sb. 180 (1989), no. 8, 1092–1118, 1151 (Russian); English transl., Math. USSR-Sb. 67 (1990), no. 2, 555–580. MR 1019483
  • 16. M. Valdivia, Projective resolution of identity in 𝐶(𝐾) spaces, Arch. Math. (Basel) 54 (1990), no. 5, 493–498. MR 1049205, 10.1007/BF01188677

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Additional Information

Wieslaw Kubis
Affiliation: Institute of Mathematics, University of Silesia, Bankowa 14, 40-007 Katowice, Poland

Vladimir Uspenskij
Affiliation: Department of Mathematics, 321 Morton Hall, Ohio University, Athens, Ohio 45701

Keywords: Valdivia compact space, open map, retract, indecomposable group
Received by editor(s): November 1, 2003
Received by editor(s) in revised form: April 11, 2004
Published electronically: February 25, 2005
Communicated by: Alan Dow
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.