First countable spaces without point-countable $\pi$-bases
Volume 196 / 2007
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).
