Zero-dimensional space

Remarks

Example: $\Q$ with the subspace topology from $\R$ is zero-dimensional. For irrational numbers $a < b$, the set $\Q \cap (a, b) = \Q \cap [a, b]$ is clopen in $\Q$. These sets form a base, since irrational numbers are dense in $\R$.

Non-example: $\R$ is not zero-dimensional. Since $\R$ is connected, its only clopen sets are $\emptyset$ and $\R$, and they do not form a base.

Some authors define zero-dimensional by Lebesgue covering dimension zero instead. The two definitions agree for separable metric spaces, but not for all spaces.