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?
When a zero-dimensional topological space is normal
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general-topology
examples-counterexamples
separation-axioms
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0See Henno Brandsma's example here: http://mathoverflow.net/questions/53300/locally-compact-hausdorff-space-that-is-not-normal – 2017-01-23
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0You can also try to find such examples in [pi-base](http://topology.jdabbs.com/search?q=%7B%22and%22%3A%5B%7B%2250%22%3Atrue%7D%2C%7B%223%22%3Atrue%7D%2C%7B%2213%22%3Afalse%7D%5D%7D). – 2017-01-23
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0where is this exercise? – 2017-01-23
1 Answers
2
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.