Borel Liftings of Borel Sets: Some Decidable and Undecidable Statements

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 Consider $X subset 2omegatimes2omega$, set $Y=pi(X)$, where $pi$ denotes the canonical projection of $2omegatimes2omega$ 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 $mathbfPi 0 2$ then $(starstar)$: ""The restriction of $pi$ to some relatively closed subset of $X$ is perfect onto $Y$"" it follows that in the present case $Y$ is also $mathbfPi 0 2$. Notice that the reverse implication $(starstar)Rightarrow(star)$ holds trivially for any $X$ and $Y$. But the implication $(star)Rightarrow (starstar)$ for an arbitrary Borel set $X subset 2omegatimes2omega$ is equivalent to the statement ""$forall alphain omegaomega, ,aleph 1$ is inaccessible in $L(alpha)$"". More precisely The authors prove that the validity of $(star)Rightarrow(starstar)$ for all $X in varSigma0 1 xi 1 $, is equivalent to ""$aleph xi aleph 1$"". However we shall show independently, that when $X$ is Borel one can, in $ZFC$, derive from $(star)$ the weaker conclusion that $Y$ is also Borel and of the same Baire class as $X$. This last result solves an old problem about compact covering mappings. In fact these results are closely related to the following general boundedness principle Lift$(X, Y)$: ""If any compact subset of $Y$ admits a continuous lifting in $X$, then $Y$ admits a continuous lifting in $X$"", where by a lifting of $Zsubset pi(X)$ in $X$ we mean a mapping on $Z$ whose graph is contained in $X$. The main result of this work will give the exact set theoretical strength of this principle depending on the descriptive complexity of $X$ and $Y$