The Countable Chain Condition for Topological Spaces

Definition: Let $(X, \tau)$ be a topological space. Then $X$ is said to satisfy the Countable Chain Condition (sometimes abbreviated ccc) if every collection of mutually disjoint open sets in $X$ is countable.

For example, $\mathbb{R}$ with the usual topology satisfies the countable chain condition. The reason comes as a consequence of the following proposition.

Proposition 1: Let $(X, \tau)$ be a topological space. If $X$ is separable then $X$ satisfies the countable chain condition.

• Proof: Let $X$ be separable. Then $X$ possesses a countable and dense subset $D$.

• Now let $\mathcal F$ be a collection of mutually disjoint open sets in $X$. Since $D$ is dense we have that $D \cap B \neq \emptyset$ for every $B \in \mathcal F$ since each $B$ is open. So for each $B \in \mathcal F$ take some $x \in D \cap B$. Let $X$ be the collection of all such $X$. Then $X$ and $\mathcal F$ share the same set size since the sets in $\mathcal F$ are disjoint.

• However, notice that $X \subseteq D$, and since $D$ is countable, $X$ is countable. Thus $\mathcal F$ is countable.

• So $X$ satisfies the countable chain condition. $\blacksquare$