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

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- by Jean Calbrix PDF
- Trans. Amer. Math. Soc.
**348**(1996), 2085-2090 Request permission

## 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.

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## Additional Information

**Jean Calbrix**- Affiliation: Laboratoire A.M.S. URA C.N.R.S. D1378, U.F.R. des Sciences, F76821 Mont Saint Aignan cedex, France
- Email: Jean.Calbrix@univ-rouen.fr
- Received by editor(s): December 3, 1993
- © Copyright 1996 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**348**(1996), 2085-2090 - MSC (1991): Primary :, 03E15, 04A15, 54-05; Secondary 54C35
- DOI: https://doi.org/10.1090/S0002-9947-96-01635-2
- MathSciNet review: 1355296