Point-countable $\pi $-bases in first countable and similar spaces
Volume 186 / 2005
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.
