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?
- $X_1$ is compact and separable
- $X_2$ is compact and separable
- $X_1 $ and $X_2 $ are both compact
- 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.