Constructible sets

Question

Is it possible to write down all the constructible sets in $\mathbf{C}$ (endowed with the Zariski topology) or some other "simple" space?

Answer 1

Well, for each complex number $z\in \mathbb{C}$, $\{z\}$ is a constructible set. So if you're going to write down all the constructible subsets, you at least need to be able to write down all the complex numbers. Depending on what you mean by "write down", this might not be possible...

But the constructible sets in $\mathbb{C}$ do enjoy a simple description: they are exactly the finite and cofinite subsets of $\mathbb{C}$ (which is to say that $(\mathbb{C},+,\cdot,0,1)$ is a strongly minimal structure). This is a consequence of the fact that any polynomial in one variable has finitely many roots.

So you can describe all constructible subsets of $\mathbb{C}$ as: $$\{X\subseteq \mathbb{C}\mid X\text{ is finite or }\mathbb{C}\setminus X\text{ is finite}\}$$