Extending Baire Property by countably many sets

by

Piotr Zakrzewski

8 pages; Proc. Amer. Math. Soc., to appear.

We prove that if ZFC is consistent so is ZFC + "for any sequence An of subsets of a Polish space (X,\tau) there exists a separable metrizable topology \tau' on X with Borel(X,\tau) subset of Borel(X,\tau'), MEAGER(X,\tau') intersected with Borel(X,\tau) equal to MEAGER(X,\tau) intersected with Borel(X,\tau), and An 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.

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