We answer several questions of V. Tkachuk [Fund. Math. 186 (2005)] by showing that ∙ there is a ZFC example of a first countable, 0-dimensional Hausdorff space with no point-countable π-base (in fact, the minimum order of a π-base of the space can be made arbitrarily large); ∙ 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; ∙ it is consistent to have a first countable, hereditarily Lindelöf regular space having uncountable π-weight and ω₁ as a caliber (of course, such a space cannot have a point-countable π-base).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm196-2-4,
author = {Istv\'an Juh\'asz and Lajos Soukup and Zolt\'an Szentmikl\'ossy},
title = {First countable spaces without point-countable $\pi$-bases},
journal = {Fundamenta Mathematicae},
volume = {193},
year = {2007},
pages = {139-149},
zbl = {1132.54003},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm196-2-4}
}
István Juhász; Lajos Soukup; Zoltán Szentmiklóssy. First countable spaces without point-countable π-bases. Fundamenta Mathematicae, Tome 193 (2007) pp. 139-149. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm196-2-4/