The following question was noted by Jan Pachl in connection with the study of Arens products and he has not received a satisfactory answer from the various experts he has asked. Let $X$ and $Y$ be compact Hausdorff spaces and let $F$ be a continuous mapping from $X$ onto $Y$. Let $A\subseteq Y$ and suppose that $F^{-1}A$ is Borel. Does it follow that $A$ is also Borel.

Certainly if $X$ and $Y$ are metric then the answer is positive; in this case both $Y\setminus A = F(X\setminus F^{-1}A)$ and $A = F(F^{-1}A)$ are analytic and hence Borel. But even for $X=Y=2^{\omega_1}$ the argument that disjoint analytic sets can be separated by Borel sets does not seem to be available.