Are two non-Turing-recognizable languages closed under union?
Let $U\subseteq \mathbb{N}$ be any unrecognizable set. Let $L_1 = \{0, 2, 4, \dots\} \cup \{2n+1\mid n\in U\}$ and $L_2 = \{1,3,5, \dots\} \cup \{2n\mid n\in U\}$. $L_1$ and&nbsp;$L_2$ are both unrecognizable but their union is $\mathbb{N}$.