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On spread and condensations

Author: A. V. Arhangelskii
Journal: Proc. Amer. Math. Soc. 124 (1996), 3519-3527
MSC (1991): Primary 54A25, 54C35, 54A35
MathSciNet review: 1353369
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Abstract: A space $X$ has a property ${\mathcal {P}}$ strictly if every finite power of $X$ has ${\mathcal {P}}$. A condensation is a one-to-one continuous mapping onto. For Tychonoff spaces, the following results are established. If the strict spread of $X$ is countable, then $X$ can be condensed onto a strictly hereditarily separable space. If $s(C_{p}(X))\leq \omega $, then $C_{p}(X)$ can be condensed onto a strictly hereditarily separable space, and therefore, every compact subspace of $C_{p}(X)$ is strictly hereditarily separable. Under $(MA+\neg CH)$, if $G$ is a topological group such that $s(C_{p}(G))\leq \omega $, then $G$ is strictly hereditarily Lindelöf and strictly hereditarily separable.

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  • 1. Arhangel'skii A.V., Topological function spaces, vol. 78, Dordrecht, Kluwer Academic Publishers, Mathematics and its Applications, 1992, (translated from Russian). MR 92i:54022
  • 2. Arhangel'skii A.V., Hereditarily Lindelöf spaces of continuous functions, Moscow Univers. Math. Bull. 44 (1989), 67-69. MR 90m:54007
  • 3. Arhangel'skii A.V., Structure and classification of topological spaces and cardinal invariants, Russian Math. Surveys 33 (1978), no. 6, 33-96. MR 80i:54005
  • 4. Arhangel'skii A.V., Left-expanded spaces, Mosc. Univ. Math. Bull. 32 (1977), 22-27. MR 58:2746
  • 5. Arhangel'skii A.V. and A. Bella, A few observations on topological spaces with small diagonal, Zbornik radova Filozofskog faculteta u Nisu Ser. Mat. 6 (1992), 211-213. CMP 1994:3
  • 6. Arhangel'skii A.V. and Fedorchuk V.V., On condensations of countably compact spaces onto compacta, Fund. Prikl. Mat. 1 (1995), 871--880. (Russian)
  • 7. Arhangel'skii A.V. and Ponomarev V.I., Fundamentals of General Topology: problems and exercises, Reidel, 1984, (translated from Russian). MR 87i:54001
  • 8. Arhangel'skii A.V. and Tkachuk V.V., Calibers and point-finite cellularity of the spaces $C_{p}(X)$, and some questions of S. Gul'ko and M Husek, Topology and Appl. 23 (1986), 65-74. MR 87h:54036
  • 9. Asanov M., Cardinal invariants of spaces of continuous functions, Modern Topology and Set Theory, Izhevskij Universitet, Izhevsk, 1979, pp. (8-12). (Russian)
  • 10. Engelking R., General Topology, PWN, Warsaw, 1977. MR 58:18316b
  • 11. Hajnal A. and I. Juhasz, On hereditarily $\alpha $-Lindelöf and $\alpha $-separable spaces, 2, Fund. Math. 81 (1974), 147-158. MR 49:1478
  • 12. Hajnal A. and I. Juhasz, A separable normal topological group need not be Lindelöf, General Topology and Appl. 6 (1976), 199-205. MR 55:4088
  • 13. Husek M., Topological spaces without $\kappa $-accessible diagonal, Comm. Math. Univ. Carol. (1977), 777-788. MR 58:24198
  • 14. Kunen K., Strong $S$ and $L$ spaces under MA, Set-theoretic Topology (G.M. Reed, ed.), Academic Press, New-York, 1977, pp. 265-268. MR 55:13362
  • 15. Negrepontis S., Banach Spaces and Topology, Handbook of Set-theoretic Topology (K. Kunen and J.E. Vaughan, eds.), North-Holland, Amsterdam, 1984, pp. (1047-1142). MR 86i:46018
  • 16. Roitman J., The spread of regular spaces, General Topology and Appl. 8 (1978), 85-91. MR 58:12909
  • 17. Roitman J., Basic $S$ and $L$, Handbook of Set-theoretic Topology (K. Kunen and J.E. Vaughan, eds.), North-Holland, Amsterdam, 1984, pp. (295-326). MR 87a:54043
  • 18. Shapirovskij B.E., On discrete subspaces of topological spaces. Weight, tightness and Souslin number, Soviet Math. Dokl. 13 (1972), 215-219. MR 45:1100
  • 19. Todorcevic S., Forcing positive partition relations, Trans. Amer. Math. Soc. 280 (1983), 703-720. MR 85d:03102
  • 20. Todorcevic S., Some Applications of $S$ and $L$ Combinatorics, Annals of the New York Academy of Sciences 705 (1993), 130-167. MR 95j:54006
  • 21. Velichko N.V., Weak topology of spaces of continuous functions, Math. Notes 30 (1981), 849-854. MR 83f:54006
  • 22. Zenor P., Hereditary $m$-separability and the hereditary $m$-Lindelöf property in product spaces and function spaces, Fundam. Math. 106 (1980), 175-180. MR 82a:54039

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Additional Information

A. V. Arhangelskii
Affiliation: Chair of General Topology and Geometry, Mech.-Math. Faculty, Moscow University, Moscow 119899, Russia (June 15–December 31); Department of Mathematics, 321 Morton Hall, Ohio University, Athens, Ohio 45701 (January 1–June 15)

Keywords: Spread, hereditary density, condensation, Lindel\"{o}f space, function spaces, topology of pointwise convergence, small diagonal, caliber
Received by editor(s): April 7, 1995
Additional Notes: The author was partially supported by NSF grant DMS-9312363.
Communicated by: Franklin D. Tall
Article copyright: © Copyright 1996 American Mathematical Society

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