Point-countable $\pi $-bases in first countable and similar spaces

Tom 186 / 2005

V. V. Tkachuk Fundamenta Mathematicae 186 (2005), 55-69 MSC: 54B10, 54C05, 54D30. DOI: 10.4064/fm186-1-4


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.


  • V. V. TkachukDepartamento de Matemáticas
    Universidad Autónoma Metropolitana
    Av. San Rafael Atlixco, 186, Col. Vicentina
    Iztapalapa, C.P. 09340, México D.F., Mexico

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