Folland in his book defines product $ \sigma $-algebra in the following way:
Let $ \{X_{\alpha} \}_{\alpha \in A} $ be an indexed collection of nonempty sets, $ X = \prod\limits_{\alpha \in A} X_{\alpha} $ and $ \pi_{a}: X \longrightarrow X_{\alpha} ~~~\textbf{coordinate maps}$. If $ M_{\alpha} $ is a $ \sigma $-algebra on $ X_{\alpha} $ for each $ \alpha$, the product $ \sigma $-algebra on $ X $ is generated by
\begin{equation} \{\pi_{\alpha}^{-1}(E_{\alpha}): E_{\alpha} \in M_{\alpha}, ~\ \alpha \in A \} \end{equation}
What do they mean in coordinate map here? Can anyone give me an example?
Coordinate maps are the projections. For instance in $\Bbb R^n$ the maps $x_j : \Bbb R^n \rightarrow \Bbb R$ defined pointwise by $x_j(a_0, ..., a_n):=a_j$ are the coordinate maps. Notice this means that the product $\sigma$-algebra is generated by the product topology.