$k$-spaces and Borel filters on the set of integers

Abstract:

We say that a countable, Hausdorff, topological space with one and only one accumulation point is a point-space. For such a space, we give several properties which are equivalent to the property of being a k-space. We study some free filters on the set of integers and we determine if the associated point-spaces are k-spaces or not. We show that the filters of Lutzer-van Mill-Pol are k-filters. We deduce that, for each countable ordinal ${\alpha \geq 2}$, there exists a free filter of true additive class ${\alpha }$ (Baire’s classification) and a free filter of true multiplicative class ${\alpha }$ for which the associated point-spaces are k-spaces but not ${\aleph _{0}}$, the existence being true in the additive case for ${\alpha =1}$. In particular, we answer negatively a question raised in J. Calbrix, C. R. Acad. Sci. Paris 305 (1987), 109–111.