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

V. V. Tkachuk¹

¹ Departamento de Matemáticas Universidad Autónoma Metropolitana Av. San Rafael Atlixco, 186, Col. Vicentina Iztapalapa, C.P. 09340, México D.F., Mexico

Fundamenta Mathematicae, Tome 186 (2005) no. 1, pp. 55-69

Résumé

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.

Détail

DOI : 10.4064/fm186-1-4

Keywords: classical result shapirovsky compact space countable tightness has point countable base look general spaces point countable bases prove particular under continuum hypothesis lindel first countable space has point countable base analyze function space has point countable base giving criterion terms topology * omega dealing point countable bases makes possible models zfc there exists space w space sense gruenhage while there exists embedding mit sigma product first countable spaces gives consistent answer twenty years old problem gruenhage