Zero-dimensional space

A space is zero-dimensional when its topology has a base of clopen sets. In the Hausdorff setting these sets separate distinct points, so the space is totally disconnected. Conversely a compact Hausdorff totally disconnected space has a clopen base, by the Šura-Bura component lemma and finite subcovers of the complement of a neighborhood.

The characteristic functions of a clopen topological basis give a topological embedding of a zero-dimensional $T_0$ topological space into a product of discrete two-point topological spaces. The separation axiom makes those functions distinguish points, and their coordinate preimages recover the original topological basis.

In a normal Hausdorff space, strong zero-dimensionality means that any disjoint closed sets can be separated by a clopen set, equivalently by disjoint clopen neighborhoods. For general Tychonoff spaces, the usual definition asks this for completely separated sets. In the normal setting Urysohn's lemma makes the formulations agree. The condition is equivalent to zero-dimensionality of the Stone-Čech compactification.

For normal Hausdorff $X$, if $\beta X$ has a clopen base, extend a Urysohn's lemma separator of two closed subsets of $X$. Their closures in $\beta X$ are separated; compactness gives a clopen separator, whose trace works in $X$. Conversely separate the traces of the low and high level sets of a continuous $f:\beta X\to [0,1]$ by clopen $U\subseteq X$. The characteristic function of $U$ extends into $\{0,1\}$ on $\beta X$. Density ensures its one level contains the zero level of $f$ and misses the one level. Applying this with $f$ separating a point from a closed set gives a clopen base for $\beta X$.