A question on $\omega_1$

Question

For the liner order space $\omega_1$, does it have countable chain condition?

(I kown it is not separable, because for any countable subset $A$ of $\omega_1$, there exists an ordinal $\kappa$ such that it is bigger for any element of $A$, so the open set $(\kappa, \omega_1)$ satisfying that $(\kappa, \omega_1)\cap A=\emptyset$.)

Thanks for help!

Answer 1

No, because $\{ \{\alpha + 1\} : \alpha < \omega_1\}$ is an uncountable set of pairwise disjoint open sets.

Note that the singleton $\{\alpha + 1\}$ can be expressed as an open interval $(\alpha, \alpha+2)$.