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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The projective $\pi$-character bounds the order of a $\pi$-base
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by István Juhász and Zoltán Szentmiklóssy PDF
Proc. Amer. Math. Soc. 136 (2008), 2979-2984 Request permission

Abstract:

All spaces below are Tychonov. We define the projective $\pi$- character $p \pi \chi (X)$ of a space $X$ as the supremum of the values $\pi \chi (Y)$ where $Y$ ranges over all (Tychonov) continuous images of $X$. Our main result says that every space $X$ has a $\pi$-base whose order is $\le p \pi \chi (X)$; that is, every point in $X$ is contained in at most $p \pi \chi (X)$-many members of the $\pi$-base. Since $p \pi \chi (X) \le t(X)$ for compact $X$, this is a significant generalization of a celebrated result of Shapirovskii.
References
  • A. V. Arhangel′skiĭ, The structure and classification of topological spaces and cardinal invariants, Uspekhi Mat. Nauk 33 (1978), no. 6(204), 29–84, 272 (Russian). MR 526012
  • A. V. Arhangel’skii, Precalibers, monolithic spaces, first countability, and homogeneity in the class of compact spaces, preprint.
  • István Juhász, Cardinal functions in topology—ten years later, 2nd ed., Mathematical Centre Tracts, vol. 123, Mathematisch Centrum, Amsterdam, 1980. MR 576927
  • István Juhász, Lajos Soukup, and Zoltán Szentmiklóssy, First countable spaces without point-countable $\pi$-bases, Fund. Math. 196 (2007), no. 2, 139–149. MR 2342624, DOI 10.4064/fm196-2-4
  • B. E. Shapirovskii, On the tightness, $\pi$-weight and related notions, Uch. Zap. Latv. Univ. 257 (1976), pp. 88–99.
  • B. È. Shapirovskiĭ, Special types of embeddings in Tychonoff cubes. Subspaces of $\Sigma$-products and cardinal invariants, Topology, Vol. II (Proc. Fourth Colloq., Budapest, 1978) Colloq. Math. Soc. János Bolyai, vol. 23, North-Holland, Amsterdam-New York, 1980, pp. 1055–1086. MR 588855
  • B. È. Šapirovskiĭ, Mappings on Tihonov cubes, Uspekhi Mat. Nauk 35 (1980), no. 3(213), 122–130 (Russian). International Topology Conference (Moscow State Univ., Moscow, 1979). MR 580628
  • B. E. Shapirovskii, Cardinal invariants in compact Hausdorff spaces, Amer. Math. Soc. Transl. 134 (1987), pp. 93–118.
  • Stevo Todorčević, Free sequences, Topology Appl. 35 (1990), no. 2-3, 235–238. MR 1058803, DOI 10.1016/0166-8641(90)90108-E
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Additional Information
  • István Juhász
  • Affiliation: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, POB 127, Budapest, H-1364 Hungary
  • Email: juhasz@renyi.hu
  • Zoltán Szentmiklóssy
  • Affiliation: Department of Analysis, Eötvös Loránt University, Pázmány Péter sétány 1/A, 1117 Budapest, Hungary
  • Email: zoli@renyi.hu
  • Received by editor(s): March 28, 2007
  • Received by editor(s) in revised form: June 17, 2007
  • Published electronically: April 2, 2008
  • Additional Notes: This research was supported by OTKA grant no. 61600.
  • Communicated by: Alexander N. Dranishnikov
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 136 (2008), 2979-2984
  • MSC (2000): Primary 54A25, 54C10, 54D70
  • DOI: https://doi.org/10.1090/S0002-9939-08-09315-5
  • MathSciNet review: 2399066