It is a classical result of Shapirovsky that any compact space of countable tightness has a point-countable π-base. We look at general spaces with point-countable π-bases and prove, in particular, that, under the Continuum Hypothesis, any Lindelöf first countable space has a point-countable π-base. We also analyze when the function space has a point-countable π -base, giving a criterion for this in terms of the topology of X when l*(X) = ω. Dealing with point-countable π-bases makes it possible to show that, in some models of ZFC, there exists a space X such that is a W-space in the sense of Gruenhage while there exists no embedding of in a Σ-product of first countable spaces. This gives a consistent answer to a twenty-years-old problem of Gruenhage.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm186-1-4,
author = {V. V. Tkachuk},
title = {Point-countable $\pi$-bases in first countable and similar spaces},
journal = {Fundamenta Mathematicae},
volume = {185},
year = {2005},
pages = {55-69},
zbl = {1080.54018},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm186-1-4}
}
V. V. Tkachuk. Point-countable π-bases in first countable and similar spaces. Fundamenta Mathematicae, Tome 185 (2005) pp. 55-69. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm186-1-4/