Why isn't the class of Turing-Recognizable languages closed under Complement?
Hint:
Suppose it was.  Let $L$ be a recognizable language and let $\overline{L}$ be its complement.  If $\overline{L}$ was recognizable as well, let $M_1$ and $M_2$ be recognizers of $L$ and $\overline{L}$, respectively.

Can you now use $M_1$ and $M_2$ to construct a decider for $L$?  What does this mean for undecidable and recognizable problems like the halting problem?