Compactness gives Second countable space??

By Tychonoff's theorem, the arbitrary product of compact spaces is compact. Take $X=\{0,1\}$ with the discrete topology and let $Y$ be the direct product of an uncountable collection of copies of $X$. Then $Y$ is compact, but $Y$ is not first countable (let alone second countable).

If $X$ is any set, the cofinite topology on $X$ is compact and any base for it has size $|X|$.