A+ CATEGORY SCIENTIFIC UNIT

First countable spaces without point-countable $\pi$-bases

Volume 196 / 2007

István Juhász, Lajos Soukup, Zoltán Szentmiklóssy Fundamenta Mathematicae 196 (2007), 139-149 MSC: 54A25, 54A35, 54D20, 54D70, 54F05. DOI: 10.4064/fm196-2-4

Abstract

We answer several questions of V. Tkachuk [Fund. Math. 186 (2005)] by showing that

$\bullet$ there is a ZFC example of a first countable, 0-dimensional Hausdorff space with no point-countable $\pi$-base (in fact, the minimum order of a $\pi$-base of the space can be made arbitrarily large);

$\bullet$ if there is a $\kappa$-Suslin line then there is a first countable GO-space of cardinality $\kappa^+$ in which the order of any $\pi$-base is at least $\kappa$;

$\bullet$ it is consistent to have a first countable, hereditarily Lindelöf regular space having uncountable $\pi$-weight and $\omega_1$ as a caliber (of course, such a space cannot have a point-countable $\pi$-base).

Authors

  • István JuhászAlfréd Rényi Institute of Mathematics
    V. Reáltanoda utca, 13–15
    H-1053 Budapest, Hungary
    e-mail
  • Lajos SoukupAlfréd Rényi Institute of Mathematics
    V. Reáltanoda utca, 13–15
    H-1053 Budapest, Hungary
    e-mail
  • Zoltán SzentmiklóssyDepartment of Analysis
    Eötvös Loránd University
    Pázmány Péter sétány 1/A
    H-1117 Budapest, Hungary
    e-mail

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