# set theory – Is the leftmost branch function \$text{UB}rightarrow omega^omega\$ Borel?

Let’s denote by $$text{UB}$$ the set of ill-founded trees over $$omega$$ having a unique branch.
Consider now the function $$b: text{UB} rightarrow omega^omega$$, that associates to each tree $$T$$ in $$text{UB}$$ its unique branch $$b_T$$. Is this function Borel?

This function is the restriction of the left-most branch function defined over ill-founded trees (possibly with multiple branches), which is $$sigma(boldsymbol{Sigma}_1^1)$$-measurable, to $$text{UB}$$. Can we in some way prove that this restriction is indeed Borel? Otherwise, do we a sort of counterexample?

Thanks