open sets of infinite produduct: $[0,1]^{\omega _1}$

Question

I am studying a paper and where assuming Continuum Hypothesis (CH). Author writes "every open set $U$ in $[0,1]^{\omega_1}$ is countable union of basic open sets.

I do not know how to prove it.

I worked using the fact that $[0,1]^{\omega_1}$ is separable and $[0,1]$ has countable basis (space $[0,1]^{\omega_1}$ is not 2-countable ) but I did not get to prove.

Thanks in advance.

Answer 1

$[0,1]^{\omega_1}$ does not have that property. Every open set $O$ does contain a countable union of basic open sets that is dense in $O$; this follows from ccc, essentially.