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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

k-spaces and Borel filters on the set of integers


Author: Jean Calbrix
Journal: Trans. Amer. Math. Soc. 348 (1996), 2085-2090
MSC (1991): Primary :, 03E15, 04A15, 54-05; Secondary 54C35
MathSciNet review: 1355296
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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

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01635-2
PII: S 0002-9947(96)01635-2
Keywords: Borel filters, point-spaces, k-spaces, $\aleph _{0}$-spaces
Received by editor(s): December 3, 1993
Article copyright: © Copyright 1996 American Mathematical Society