How can I show that this set is (or not is) a Borel set?

Question

Suppose we have two Polish spaces $X$ and $Y$ and two Borel injective functions $f\colon X\to Y$ and $g\colon Y\to X$. Consider the following map $$\Phi: \mathcal P(X)\to \mathcal P(X);\ Z\mapsto X\setminus g(Y\setminus f(Z)).$$ It has a fixed point in $\bigcup\{Z\subseteq X\mid Z\subseteq\Phi(Z)\}\in\mathcal P(X)$; is this set a Borel set of $X$? If it is, how can I show that?

Answer 1

It can be proved that the operator $\Phi$ is continuous, meaning that the fixpoint can be obtained as the intersection (the “limit”) of the iterates $\Phi^{(n)}(X)$. Since injective Borel maps between Polish spaces send Borel sets to Borel sets, each iterate is Borel and thus the intersection also is.