I am trying to work through some definitions in algebraic geometry using some easy examples and I would like to have a confirmation or disproof of the following:
The locally constructible subsets of $\mathrm{Spec} \mathbb{Z}$ are exactly the open and the closed subsets.
My reasoning:
- Every subset of $\mathrm{Spec} \mathbb{Z}$ is quasi-compact, hence also retrocompact
- The constructible subsets are therefore exactly the open and closed subsets
- The same is true for the locally contructible subsets
I have skipped the reasonings in between, of course, but are the above statements correct?