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Suppose $L_1\preccurlyeq_T L_2$, and $L_1$ is unrecognizable, can $L_2$ be recognizable?

With decidability, if $L_1$ is undecidable, then $L_2$ is undecidable, because $L_1$ is the “easier” question. Would this apply to recognizability, too?

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Any set $A$ Turing-reduces to its complement $A^c$.

Hence, the non-recognizable set $K^c$ Turing-reduces to the recognizable set $K=(K^c)^c$, where $K$ is the Kleene set (or, if you prefer, your favorite variation on the halting problem).

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