When a zero-dimensional topological space is normal

Question

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?

Answer 1

No this is false, some examples:

1. https://mathoverflow.net/a/53301/2060 describes the deleted Tychonov plank.

2. https://mathoverflow.net/a/56805/2060 describes the rational sequence topology

3. https://math.stackexchange.com/a/462029/4280 descibres Mrowka $\Psi$-space

4. 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.

Answer 2

I'm a few years late, but the missing assumption in your exercise might be that of second countability of the space. It is indeed true that second countable hausdorff zero dimensional spaces are normal, because zero dimensional implies (completely) regular and regular plus second countable implies normal.

Showing that if $X$ is Hausdorff, second countable and zero dimensional, then for every pair of closed sets $C_1$ and $C_2$ in $X$ there is a clopen set $B$ with $C_1\subseteq B$ and $C_2\cap B=\varnothing$ is exercise 7.2 in Kechris's book Classical Descriptive Set Theory