Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

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

Tom 186 / 2005

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

Streszczenie

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.

Autorzy

• V. V. TkachukDepartamento de Matemáticas