= Strong zero-dimensionality and Stone-Čech compactification
{title2=$X\text{ strongly zero-dimensional}\iff\beta X\text{ zero-dimensional}$}
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 $\beta X$.
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