2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/126390We 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).General Topology54A25 (Primary), 54A35, 54D20, 54D70, 54F05 (Secondary)text