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First countable spaces without point-countable $π$-base

Istvan Juhasz·Lajos Soukup·Zoltan Szentmiklossy
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Abstract

We answer several questions of V. Tkačuk from [Point-countable $π$-bases in first countable and similar spaces, Fund. Math. 186 (2005), pp. 55--69.] by showing that (1) there is a ZFC example of a first countable, 0-dimensional Hausdorff space with no point-countable $π$-base (in fact, the order of any $π$-base of the space is at least $\aleph_ω$); (2) if there is a $κ$-Suslin line then there is a first countable GO space of cardinality $κ^+$ in which the order of any $π$-base is at least $κ$; (3) it is consistent to have a first countable, hereditarily Lindel\" of regular space having uncountable $π$-weight and $ω_1$ as a caliber (of course, such a space cannot have a point-countable $π$-base).

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Istvan Juhasz, Lajos Soukup, Zoltan Szentmiklossy. 2007-03-25. First countable spaces without point-countable $π$-base. https://arxiv.org/abs/math/0703728

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