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

István Juhász, Lajos Soukup and Zoltán Szentmiklóssy

We answer several questions of V. Tkacuk 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 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).