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Is there any good strategy of examining whether a given algebraic set is closed or dense in the metric topology of Euclidean space? For example suppose we are in $\mathbb{R}^3$ and consider the set $V=\left\{(x,y,z):x^2-zy=0 \right\}$. How can we see whether $V$ is closed or dense or dismiss these possibilities? The topology of interest is the one induced by the Euclidean distance.

Edit: I am particularly interested in determining whether the intersection of an irreducible variety in the Zariski topology with the unit sphere in the Euclidean topology is a compact set in the Euclidean topology.

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    $\begingroup$ Consider the function $f:\mathbb{R}^3\rightarrow \mathbb{R}$ that sends $(x,y,x)$ to $x^2-yz$. $f$ is clearly continuous. Since $\{0\}$ is closed in $\mathbb{R}$, therefore $V=f^{-1}[\{0\}]$ is closed. Am I missing something ? $\endgroup$ – Amr May 6 '13 at 22:18
  • $\begingroup$ @Amr: Fantastic, no, i was missing the obvious. $\endgroup$ – Manos May 6 '13 at 22:23
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The set of points in $\mathbb R^n=\{(x_1,x_2,\ldots,x_n):x_k\in\mathbb R\text{ for each }k\}$ where some polynomial $p(x_1,x_2,\ldots,x_n)$ vanishes is always closed, because polynomials are continuous (relative to the metric topology), resulting from continuity of addition and multiplication.

If a set is closed and dense, then it must be equal to the whole space. Therefore, to show that such a set is not dense, it suffices to show that it is not equal to the whole space, i.e., that there exists a point $(x_1,x_2,\ldots,x_n)$ such that $p(x_1,x_2,\ldots,x_n)\neq 0$.

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