There're two definitions of Borel pointclass $\Sigma_2^0$. One is that $P \in \Sigma_2^0$, iff there exists a closed subset $F$ of $\chi \times \omega$ such that for all $x \in P $,there exists $t \in \omega$ and $(x,t) \in F$.
Another is that $P \in \Sigma_2^0$, iff $P = \bigcup_{i=0}^{\infty}F_i$ in which each $F_i$ is closed.
$\chi$ is a finite product space $\Pi_{1 \leq i \leq n}X_i$, $X_i$ might take $\omega$, $\Bbb R$, $\omega^{\omega}$, $2^{\omega}.$
How to show these two definitions are equivalent?
If I understand correctly what the first definition is supposed to be, it will imply the second definition because projecting along a coordinate indexed by $\omega$ amounts to taking a countable union. Proving the converse will amount to observing that because $\omega$ is discrete, any $\omega$-sequence of closed subsets of $\chi$ corresponds to a closed subset of $\omega \times \chi$.