Corollary 2

Tag 0BPP, permanent. Cite it as [Bourbaki, Tag 0BPP].

General Topology, chapter IV, § 2. Fundamental topological properties of the real line

The only compact connected subsets of $\mathbf{R}$ are the bounded closed intervals.

By Theorem 4, a subset of $\mathbf{R}$ which does not contain any interval consisting of more than one point is totally disconnected; this is true, for example, of the set $\mathbf{Q}$ of rational numbers, since the set $C\mathbf{Q}$ of irrational numbers is dense in $\mathbf{R}$ .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Theorem 4