Abstract
We answer several questions of V. Tkaycuk 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 ℵ!); (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 Lindelof regular space having uncountable �-weight and !1 as a caliber (of course, such a space cannot have a point-countable �- base).
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