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Point-countable π-bases in first countable and similar spaces

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Tkachuk V.

Fundamenta Mathematicae

2005

tom 186

nr 1

55-69

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

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1 Fundamenta Mathematicae

1 Tkachuk V.

1 2005

Wyniki wyszukiwania

Point-countable π-bases in first countable and similar spaces