Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Fundamenta Mathematicae / All issues

Abstract

It is a classical result of Shapirovsky that any compact space of countable tightness has a point-countable $\pi$-base. We look at general spaces with point-countable $\pi$-bases and prove, in particular, that, under the Continuum Hypothesis, any Lindelöf first countable space has a point-countable $\pi$-base. We also analyze when the function space $C_{\rm p}(X)$ has a point-countable $\pi$-base, giving a criterion for this in terms of the topology of $X$ when $l^*(X)=\omega$. Dealing with point-countable $\pi$-bases makes it possible to show that, in some models of ZFC, there exists a space $X$ such that $C_{\rm p}(X)$ is a $W$-space in the sense of Gruenhage while there exists no embedding of $C_{\rm p}(X)$ in a ${\mit \Sigma }$-product of first countable spaces. This gives a consistent answer to a twenty-years-old problem of Gruenhage.