Proposition 2

Every non-empty open set in $\mathbf{R}$ is the union of a countable family of mutually disjoint open intervals.

Let $A$ be a non-empty open set in $\mathbf{R}$. Since $\mathbf{R}$ is locally connected, every component of $A$ is a connected open set (Chapter I, § 11, no. 6, Proposition 11) and therefore an open interval by Theorem 4. Any two of these open intervals are disjoint; on the other hand, each of them contains a rational number; hence the set of these intervals has a power less than or equal to that of $\mathbf{Q}$, i.e. is countable.

It follows that every closed set in $\mathbf{R}$ is the complement of the union of a (finite or infinite) sequence $(I_n)$ of mutually disjoint open intervals. These intervals are said to be contiguous to the closed set under consideration. Conversely, given such a sequence of intervals, the complement of their union is a closed set to which these intervals are contiguous.