Totally disconnected space

Definition

A topological space $X$ is totally disconnected when each connected subset of $X$ has at most one point.

Remarks

Equivalently, each connected component of $X$ is a single point. Each discrete space is totally disconnected. So are $\mathbb{Q}$ and $\mathbb{R} \setminus \mathbb{Q}$, as subspaces of $\mathbb{R}$. The Cantor set is totally disconnected, although it is compact and has no isolated points.

Explanations

L1 human

A totally disconnected space falls apart into single points: no subset with two or more points is in one piece. This does not mean that the space is discrete. The rationals are a good example.