Separable space and countable How do I show that every collection of disjoint open sets in a separable space must be countable? 
I am studying separable from this book and this was stated, and the prove was left to the reader. I am still trying to understand the material, so I have some problem with this, can anyone give me a light, please?
 A: Can you find an injection from the collection $\{U_i\mid i\in I\}$ of disjoint open sets to the countable dense subset $D$? Then the cardinality of the collection cannot be larger than the cardinality of $D$.
A: Since your space is separable there is a countable subset S which is dense. If your collection of open subsets is uncountable you could choose an element of S from each of these which would contradict the fact that S is countable.
A: This may be helpful for your question.
Let us recall that a space $X$ is said to be CCC, i.e., countable chain condition if $X$ satisfies the condition that every collection of disjoint open sets in a space must be countable. Note that

Every separable space is CCC. The converse is not true.

For example:

Let $X$ be a space with $|X|= \aleph_1$, let $\tau_X= \lbrace U: X\setminus U \text{ is countable } \rbrace$. This space is CCC, but not separable.
Proof: $X$ is not separable: for any countable set $A \subset X$, clearly, $U=X\setminus A$ is open and $U \cap A=\emptyset$.
$X$ is CCC: if $X$ has uncountable disjoint open sets $\lbrace \cal U_\xi: \xi \in \aleph_1\rbrace$. Pick one open set, for example, $U_0$. Because $X \setminus U_0$ is uncountable, it is a contradiction with $U_0$ is open.

