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Consider the Zariski Topology on $\mathbb{C}^n.$ Then is it true that for every non-empty Zariski open set $U,$ $U \cap \mathbb{R}^n$ is open dense in $\mathbb{R}^n$?

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    you have to precise the topology used in $\mathbb{R}^n$.2012-04-09
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    The topology on $\mathbb{R}^n$ is the usual topology.2012-04-09
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    if $U$ is open for the Zariski topology in $\mathbb{C}^n$, can we conclude that $U$ is open for the usual topology in $\mathbb{C}^n$ ?2012-04-09
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    Yes. It is not difficult to see that a Zariski closed is closed in usual topology. Hence a Zariski open set is open in usual topology.2012-04-09
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    if $U$ a Zariski open set, is $U$ bounded ?2012-04-09
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    @Matrix: No. Consider $U_x = \mathbb{R}^2 \backslash Z(x) $, the complement of a line in $\mathbb{R}^2$. Not bounded.2012-04-09

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