**István Juhász,
Lajos Soukup and Zoltán Szentmiklóssy**

- 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 );
- 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 ;
- 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).

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**2000 Mathematics Subject Classification**:
Primary: , Secondary: