One of the aims of this work is to investigate some natural properties of Borel sets which are undecidable in $ZFC$. The authors' starting point is the following elementary, though non-trivial result: Consider $X subset 2omega imes2omega$, set $Y=pi(X)$, where $pi$ denotes the canonical projection of $2omega imes2omega$ onto the first factor, and suppose that $(star)$: Any compact subset of $Y$ is the projection of some compact subset of $X$. If moreover $X$ is $mathbf{Pi 0 2$ then $(starstar)$: The restriction of $pi$ to some relatively closed subset of $X$ is perfect onto $Y$ it follows...