TITLE: How to prove a space is not separable?
QUESTION [5 upvotes]: I have a general question on how to prove a space is not separable.
I read some posts on this site and it seems like it suffices to find an uncountable family of pairwise disjoint open sets to prove a space is not separable. (here: The space of bounded continuous functions are not separable)
Why an uncountable family of pairwise disjoint open sets is enough?

REPLY [9 votes]: A separable space $X$ is a space with a countable dense subset $S$. Label the elements of the $S$ with $x_i$. Since $S$ is dense, $\bar{S}=X$. Thus any open set $U$ must contain some $x_i$. However, if $X$ contains uncountably many disjoint opens, this is not possible, as there must be some $U_\alpha$ not containing any of the $x_i$.