Is every first countable profinite group actually second countable?



Is every first countable profinite group, second countable?
Although this question has been answered already by Masked Avenger, it seems like a good idea for me to explain why every first-countable Hausdorff group is metrizable. If $G$ is a Hausdorff topological group and $U$ is an open neighborhood of the identity $e$, then let $R\_{U}$ be the binary relation on $G$ where $(x,y)\in R\_{U}$ iff $xy^{-1}\in U$. Then $G$ becomes a Hausdorff uniform space with a uniformity generated by the relations $R\_{U}$ where $U$ ranges over all open neighborhoods of the identity $e$. Furthermore, it is well known that a uniform space $(X,\mathcal{U})$ is induced my a metric if and only if $\mathcal{U}$ is generated by countably many relations (a proof of this fact is given in [this answer](https://mathoverflow.net/questions/123160/a-uniformity-with-a-countable-base-is-a-pseudometric-uniformity)). Therefore, if a $G$ is a first countable Hausdorff topological group, then the uniformity generated by the relations of the form $R\_{U}$ is metrizable. Therefore, we conclude that every Hausdorff first countable topological group is metrizable.


