totally disconnected space

A topological space $X$ is called totally disconnected if all connected components of $X$ are singletons.

Examples of totally disconnected spaces include:

• Every discrete space,

• Cantor set,

• the set of rational numbers $\mathbb{Q}$ endowed with the subspace topology induced by $\mathbb{Q}\hookrightarrow ℝ$ and,

• the set of irrational numbers $ℝ\setminus \mathbb{Q}$ again with the subspace topology.