1-st countability of a uncountable topological product Let $X = \displaystyle\prod_{\lambda \in \Lambda} X_{\lambda}$, where each $X_\lambda$ is $T_2$ and has at least two points. Prove that if $\Lambda$ is uncountable then $X$ is not 1st-countable.
I dont even know how to start proving. Maybe in the opposite direction, $X$ 1st-countable $\implies$ $\Lambda$ countable...
I know I have to use the $T_2$ assumption, but I dont know wheter its as simple as noticing that X is  also $T_2$, or do I have to do something with the projections to ceratin $\lambda$, and taking into account that the projections are continious.
Thanks

Hint: Membership in a countable family of open sets depends only on countably many coordinates, as is being a superset of a member of such a family. Being a neighbourhood of a point in the product depends on $\lvert \Lambda\rvert$ many coordinates (due to each $X_\lambda$ being nontrivial).
In fact, if you choose your point carefully, all you need about $X_\lambda$ is for each to be nontrivial (as in nontrivial topology), so that there's an $x_\lambda\in X_\lambda$ with a neighbourhood distinct from $X_\lambda$ itself.
