Let $X$ denote the product of countably many copies of [$0,1$] . we let $X_1$ denote the set $X$ equipped with the box topology and let$X_2$ denote the se $X$ equipped with the product topology. Then
(1) $X_1$ is compact and separable.
(2) $X_2$ is compact and separable.
(3) $X_1$ and $X_2$ are both compact
(4) Neither $X_1$ nor $X_2$ is separable
Box topology is not compact nor separable but product topology is both. [0,1] is compact metric space so separable. So 1 is correct.but not sure about the others.can anybody help me .thanks.