Zero Locus of Vanishing Ideal is Closure
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Theorem
Let $k$ be a field.
Let $n \ge 1$ be a natural number.
Let $\mathbb A^n_k$ be the standard affine space over $k$.
Let $\mathbb A^n_k$ be equipped by Zariski topology.
Let $S \subseteq \mathbb A^n_k$.
Then:
- $\map V {\map I S} = S^-$
where:
- $\map I \cdot$ denotes the vanishing ideal
- $\map V \cdot$ denotes the zero locus
- $S^-$ is the closure of $S$
Proof
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