Strong zero-dimensionality

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 $\beta X$.