Let $\displaystyle \{X_i\}_{i \in I}$ a family of topological spaces. Prove that the product space $\displaystyle \prod X_i$ is first-countable if and only if each $\displaystyle X_i$ are first-countable, and every space $\displaystyle X_i$, except for a countable quantity, are indiscrete.

I have no idea... Maybe something for the converse, but probably I´m wrong...