Let $X,Y$ be Polish spaces, and $B\subseteq X \times Y$ a Borel subset. The projection $B_X$ is not necessarily Borel in $X$. I have seen a few sufficient conditions for the projection to be Borel, namely if the sections $B_x$ are: 1. singletons, 2. nonmeager, 3. open, 4. compact, 5. $\sigma$-compact.
I would like to know whether there is an equivalent condition which states exactly which Borel sets $B$ have Borel projections (does not necessarily have to be related to the sections).
But any other sufficient conditions would be super useful as well!
Many thanks in advance!
