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