TITLE: Zero-dimensional but not Hausdorff
QUESTION [5 upvotes]: 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?

REPLY [6 votes]: 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$.