Extending Baire Property by countably many sets

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.