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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.

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    $\begingroup$ 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 $\mathbb{R}^{2 n}$. $\endgroup$ – Zhen Lin Jan 13 '15 at 10:42
  • $\begingroup$ @ZhenLin: That seems to be right. Thank you! $\endgroup$ – Dune Jan 13 '15 at 14:27

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