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Nicely generated and chaotic ideals

Author: Jörg Brendle
Journal: Proc. Amer. Math. Soc. 124 (1996), 2533-2538
MSC (1991): Primary 03E05, 03E40, 28A05, 54H05
MathSciNet review: 1322913
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Abstract: We show that if the real line is the disjoint union of $\kappa $ meager sets such that every meager set is contained in a countable union of them, then $\kappa = \omega _1$. This answers a question addressed by Jacek Cicho\'{n}. We also prove two theorems saying roughly that any attempt to produce the isomorphism type of the meager ideal in the Cohen real and the random real extensions must fail. All our results hold for meager replaced by null, as well.

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Jörg Brendle
Affiliation: Mathematisches Institut der Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
Address at time of publication: Department of Mathematics, Bradley Hall, Dartmouth College, Hanover, New Hampshire 03755

Keywords: Meager sets, null sets, Cohen reals, random reals
Received by editor(s): July 15, 1994
Received by editor(s) in revised form: February 23, 1995
Additional Notes: Supported by DFG–grant Nr. Br 1420/1–1.
Communicated by: Andreas R. Blass
Article copyright: © Copyright 1996 American Mathematical Society

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