Arithmetical Hierarchy, show $\Sigma_1$ is Turing recognizable
Suppose that $L = \{x \mid \exists y R(x,y)\}$, where $R(x,y)$ is a decidable predicate.

To recognize $L$, given an input $x$, enumerate all possible $y$, and check whether $R(x,y)$ holds for at least one of them. This will halt iff $x \in L$.