# First countable spaces without point-countable -bases

István Juhász, Lajos Soukup and Zoltán Szentmiklóssy

We answer several questions of V. Tkacuk from [Point-countable -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 -base (in fact, the order of any -base of the space is at least );

2. if there is a -Suslin line then there is a first countable GO space of cardinality in which the order of any -base is at least ;

3. it is consistent to have a first countable, hereditarily Lindelöf regular space having uncountable -weight and as a caliber (of course, such a space cannot have a point-countable -base).