A topological space is called zero-dimensional whenever it has a clopen basis for open sets. There is an exercise that states every Hausdorff zero-dimensional space is normal, but I think it is false and there is a missed assumption in this exercise, am I right?
2026-03-26 12:32:27.1774528347
When a zero-dimensional topological space is normal
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No this is false, some examples:
https://mathoverflow.net/a/53301/2060 describes the deleted Tychonov plank.
https://mathoverflow.net/a/56805/2060 describes the rational sequence topology
https://math.stackexchange.com/a/462029/4280 descibres Mrowka $\Psi$-space
https://math.stackexchange.com/a/170740/4280 gives a proof of non-normality of the Sorgenfrey square $S \times S$, where $S$ is the reals in the lower limit topology (generated by the clopen sets $[a,b)$, the square thus is also zero-dimensional).
All of these are zero-dimensional Hausdorff and not normal. The first 3 are also locally compact. 2,3 and 4 are also separable (and first countable). So some conditions extra need not be enough to get normality.