Im slightly struggling to visualise the concept here. On the reals or the complex plane, the open ball with euclidean topology is not an affine algebraic variety. How is there no collection of polynomials that has it as its zero locus?
Why is the Open ball in Euclidean topology not an affine algebraic variety?
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algebraic-geometry
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1actually i Think i may see it now – 2017-02-27
1 Answers
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polynomials vanish on closed sets, since they are continuous, and $0 \in \mathbb C$ is closed, so that $P^{-1}(\{0\})$ is closed for any polynomial $P$.
Because $k[x_1,..,x_n]$ is a noetherian ring, it's enough to consider a finite collection of polynomials, say $f_{i}$ for $i \in I$, where $I$ is finite. Because $V(\bigcup_{i \in I} f_i)=\bigcap_{i \in I} V(f_i)$ is the intersection of a finite number of closed sets, it is again closed.
Because of this, the open ball cannot be the vanishing locus for some finite collection of polynomials.