Connected sets in $\Bbb R$ and open subsets in $\Bbb R$

Question

In this particular post : Any open subset of $\Bbb R$ is a at most countable union of disjoint open intervals. [Collecting Proofs] , it is shown that any open subset of $\Bbb R$ is a countable union of disjoint open intervals. I am getting confused. I know that connected sets in $\Bbb R$ are intervals, say $(a,b) \subset \Bbb R$ and that the definition of connected says that a connected set cannot be decomposed into two disjoint open sets. So how can every open subset be a disjoint countable union of open sets?

Answer 1

There is no contradiction: all open subsets of $\mathbb{R}$ are at most countable unions of open intervals (if we want a disjoint union, we should also include open segments in this). All connected open subset of $\mathbb{R}$ are open intervals, open segments (like $(a,\infty)$ and $(-\infty,b)$) and $\mathbb{R}$ and $\emptyset$ itself. The connected ones are the building blocks for all open sets (this is what locally connected means).

Answer 2

At most countable may as well mean one -- this is the case for the open intervals. Then there is no contradiction with the conectedness.