Let A be a distributive lattice and SpecA the set of prime ideals of A. The opens of SpecA are of the form $\{P|I\not\subseteq P\}$ for I an ideal of A.
We know that SpecA is Hausdorff iff A is a Boolean algebra.
The zero sets $Z(X)$ of a topological space $X$ form a distributive lattice and $SpecZ(X)$ is Hausdorff.
But $Z(X)$ is not a Boolean algebra in general. Where am I wrong?