Definition 5

A topological space is said to be zero-dimensional if it is Hausdorff and if every point has a fundamental system of neighbourhoods which are both open and closed.

Every zero-dimensional space $X$ is totally disconnected; for the component of a point $x$ is contained in all the sets containing $x$ which are both open and closed (Chapter I, § II, no. 5), and the intersection of these sets is just $\{ x \}$ if $X$ is zero-dimensional.

Conversely, a locally compact totally disconnected space is zero-dimensional (Chapter II, § 4, no. 4, Proposition 6); but there exist totally disconnected metrizable spaces which are not zero-dimensional [Exercise 3 b)].

Every subspace of a zero-dimensional space is zero-dimensional, and topological sums and products of zero-dimensional spaces are zero-dimensional.