Let's call a space zero-dimensional if it has a basis of clopen sets, and is $T_0$. Is there a zero-dimensional space that is not Hausdorff?
2026-03-26 22:50:30.1774565430
Zero-dimensional but not Hausdorff
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No. Let $x\neq y$ and let be $ U$ a clopen neighbourhood of $x$ not containing $y$. Then the complement of $U$ is a clopen neighbourhood of $y$, disjoint from $ U$.