Let $X$ be a compact metric space and $Y\subset X$ be a compact set. Assume that $f\_1, f\_2: Y \to \mathbb{P}\mathbb{R}^2$ are continuous functions. Let $N \subset \mathbb{P}\mathbb{R}^2$ be a countable set.


### Question


Is there a Borel measurable function $T:f\_1(Y) \to f\_2(Y)$ such that if $x \in Y$ and $f\_1(x) \notin N$, then $T(f\_1(x))=f\_2(x)$?



Existence of a Borel measurable function
Let $Y=[-1,1]$. We can also view $Y$ by homeomorphic embedding and abuse of notation as a subset of $\mathbb{P}\mathbb{R}^2$. For all $x\in Y$, let $f\_1(x)=|x|$ and $f\_2(x)=x$. For all but the countably many $x$ such that $f\_1(x)=|x|\in N$, one would need $$x=f\_2(x)=T(f\_1(x))=T(|x|)=T(|-x|)=T(f\_1(-x))=f\_2(-x)=-x,$$
which is impossible.


