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Extending Baire Property by countably many sets

Author: Piotr Zakrzewski
Journal: Proc. Amer. Math. Soc. 129 (2001), 271-278
MSC (2000): Primary 03E35, 54E52; Secondary 28A05
Published electronically: June 14, 2000
MathSciNet review: 1695095
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Abstract: We prove that if ZFC is consistent so is ZFC + ``for any sequence $(A_{n})$ of subsets of a Polish space $\langle X,\tau\rangle $ there exists a separable metrizable topology $\tau'$ on $X$ with $\mathbf{B}(X,\tau)\subseteq\mathbf{B}(X,\tau')$, $\operatorname{MGR}(X,\tau')\cap\mathbf{B}(X,\tau)=\operatorname{MGR} (X,\tau)\cap\mathbf{B}(X,\tau)$ and $A_{n}$ Borel in $\tau'$ for all $n$.'' This is a category analogue of a theorem of Carlson on the possibility of extending Lebesgue measure to any countable collection of sets. A uniform argument is presented, which gives a new proof of the latter as well.

Some consequences of these extension properties are also studied.

References [Enhancements On Off] (What's this?)

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Additional Information

Piotr Zakrzewski
Affiliation: Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland

Keywords: Measure and category, Borel sets, Baire Property, $\sigma$-algebra, $\sigma$-ideal
Received by editor(s): July 10, 1998
Received by editor(s) in revised form: March 16, 1999
Published electronically: June 14, 2000
Additional Notes: The author was partially supported by KBN grant 2 P03A 047 09 and by the Alexander von Humboldt Foundation
Communicated by: Carl G. Jockusch, Jr.
Article copyright: © Copyright 2000 American Mathematical Society