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

Abstract

We answer several questions of V. Tkačuk from [Point-countable $\pi$-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 $\pi$-base (in fact, the order of any $\pi$-base of the space is at least ${\aleph }_{\omega }$); (2) 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 $\kappa$; (3) it is consistent to have a first countable, hereditarily Lindel\" of regular space having uncountable $\pi$-weight and ${\omega }_1$ as a caliber (of course, such a space cannot have a point-countable $\pi$-base).