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

István Juhász¹ ; Lajos Soukup¹ ; Zoltán Szentmiklóssy²

¹ Alfréd Rényi Institute of Mathematics V. Reáltanoda utca, 13–15 H-1053 Budapest, Hungary
² Department of Analysis Eötvös Loránd University Pázmány Péter sétány 1/A H-1117 Budapest, Hungary

Fundamenta Mathematicae, Tome 196 (2007) no. 2, pp. 139-149.

Résumé

We answer several questions of V. Tkachuk [Fund. Math. 186 (2005)] by showing that$\bullet$ there is a ZFC example of a first countable, 0-dimensional Hausdorff space with no point-countable $\pi$-base (in fact, the minimum order of a $\pi$-base of the space can be made arbitrarily large); $\bullet$ 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 $\kappa$;$\bullet$ 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).

Détail

DOI : 10.4064/fm196-2-4

Keywords: answer several questions tkachuk fund math showing bullet there zfc example first countable dimensional hausdorff space point countable pi base minimum order pi base space made arbitrarily large bullet there kappa suslin line there first countable go space cardinality kappa which order pi base least kappa bullet consistent have first countable hereditarily lindel regular space having uncountable pi weight omega caliber course space cannot have point countable pi base