Suppose $S$ is a collection of pairwise disjoint open sets in $\mathbb{R}^2$
$S$ can not be finite
S can not be countably infinite.
S can not be uncountably infinite
S is empty.
1 is wrong I can take any finite no of disjoint open sets by housdorff property I can find right?
2 is also wrong I can take points from $\mathbb{N}\times \mathbb{N}$ and seperate them by those pairwise disjoint open sets so here S is countably infinite,
3 is also wrong I will do the same thing by putting $\mathbb{Q}^c\times \mathbb{Q}^c$
so 4 is right. Is my arguments are ok?