# Properties of a modified Zariski topology

For any set $I \subseteq \mathbb{C}[X_1, \dots, X_n, Y_1, \dots, Y_n]$ of polynomials let us define $$V'(I) := \{ x \in \mathbb{C}^n : f(x,\overline{x}) = 0 \text{ for all } f \in I \}.$$ In analogy to the Zariski topology, these sets also form the closed sets of a topology on $\mathbb{C}^n$.

Is this topology well known? Does it have a name? Are there references, where I can look up its properties? I am particularly interested in the question, whether all nonempty open sets in this topology are dense in the euclidean topology.

• Well, if you have both $z$ and $\bar{z}$, then you can separate out the real and imaginary parts of $z$. So I would guess that this is the same as the Zariski topology on $\mathbb{R}^{2 n}$. – Zhen Lin Jan 13 '15 at 10:42
• @ZhenLin: That seems to be right. Thank you! – Dune Jan 13 '15 at 14:27