Let $(X_{\mu}, \tau_{\mu})_{\mu \in M}$ a family of topological spaces, if $x \in (\Pi X_{\mu}, \tau)$, $\tau$ is product topological. Prove $C(x) = \Pi C(x_{\mu})$.
We define $C(x)$ as union of all subspaces connected of $(X,\tau)$ which has $x$. But how could you define $\Pi C(x_{\mu})$?