= Zariski closure
{c}
{title2=$\overline S^{\mathrm{Zar}}=V(I(S))$}
The <Zariski closure> of a <subset> $S$ of an <algebraic variety> or a <scheme> is the smallest <Zariski-closed subset> containing $S$. For $S\subseteq k^n$, define the <ideal> $I(S)$ of all <polynomials> vanishing on $S$. Then
$$
\overline S^{\mathrm{Zar}}=V(I(S)).
$$
This description follows because an intersection of polynomial zero sets is again the common zero set of their equations.<The Stacks project>[Algebraic geometry, affine algebraic sets and ideals; topology of spectra]{ref}
Every infinite subset of the complex <affine line> is <Zariski dense>: a nonzero one-variable <polynomial> has only finitely many <roots of a polynomial>. Thus $S=\{1,1/2,1/3,\ldots\}$ has Zariski closure all of $\mathbb C$, although its closure in the ordinary <Euclidean topology> is only $S\cup\{0\}$. In <algebraic geometry>, taking a closure retains all polynomial consequences, rather than all ordinary sequential limits.
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