# A question about compactness and separability with respect to box and product topology

Let $X$ denote the product of countably many copies of $[0,1]$. Let $X_1$ denote the set $X$ with the box topology and $X_2$ denote the set $X$ equipped with the product topology.Then which ones are correct?

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

I know a topological space is compact if every open cover has a finite subcover and separable means the space has a countable dense subset.I know $[0,1]$ is compact and finite product of compact spaces is compact, but for infinite case I don't know anything. Please help. Thank you.

• No infinite box product of nondegenerate Hausdorff spaces is separable. If $X_i$, $i ∈ I$, are the spaces and $U_{i, 0}, U_{i, 1}$ are nonempty disjoint open in $X_i$, then $\{∏_{i ∈ I} U_{i, f(i)}: f ∈ \{0, 1\}^I\}$ is a family of at least continuum nonempty disjoint sets open in box topology.