Let $I\neq\emptyset$ numerable and $(X_\alpha,\tau_\alpha)$ a family of topological spaces. Prove the following.
$\displaystyle\prod X_\alpha$ is first-countable if and only if $X_\alpha$ is first-countable, $\forall \alpha\in I.$
$\displaystyle\prod X_\alpha$ is second-countable if and only if $X_\alpha$ is second-countable, $\forall \alpha\in I.$
All I have are the definitions and I do not know how to proceed to do the proofs.
Could you give me the idea of the proof for 1. and 2. please ? ? ?
Note that I am not asking for the entire proof since I know this site does not work like that.
$(X,\tau)$ is first-countable if $\forall x\in X,$ has a countable local basis of neighborhoods.
$(X,\tau)$ is second-countable if $\tau$ has a countable basis, i.e. if $\exists\beta\subset\tau:\beta $ is countable and basis for $\tau.$