# Is there an undecidable language that is mapping reducible to its complement?

Is there an undecidable language A that is mapping reducible to its complement?

If it is possible, since A is an undecidable language, so A's complement must also be an undecidable language. But i don't know whether undecidable languages is closed under complement or not.

Yes, it is possible. Enumerate the set of all possible Turing machines and let $$H$$ (resp. $$\overline{H}$$) be the set of indices of the Turing machines that halt (resp. do not halt) on empty input.
Let $$L = \{ \langle T, 1 \rangle \, : T \in H \} \cup \{ \langle T, 0 \rangle \, : T \in \overline{H} \}$$.
Clearly $$L$$ is not decidable but it is possible to reduce $$L$$ to $$\overline{L}$$ since $$\langle T, r \rangle \in L \iff \langle T, 1-r \rangle \in \overline{L}$$.