Extending Baire Property by countably many sets

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.