Zariski topology

The Zariski topology on an affine scheme $\mathrm{Spec}A$ has closed sets $V(I)=\{\mathfrak{p}:I\subseteq \mathfrak{p}\}$ for ideals $I\subseteq A$. Its basic open sets are the principal open subschemes $D(f)$.

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 $S$. ThenThis description follows because an intersection of polynomial zero sets is again the common zero set of their equations.^[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 $S=\{1,1/2,1/3,\dots \}$ 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.

A Zariski cover of a scheme is a family of open immersions whose images cover its underlying topological space. Such families define the Zariski topology as a Grothendieck topology on the category of schemes; a cover need not be finite.

A subset of an algebraic variety is Zariski dense when its closure in the Zariski topology is the whole variety. For an affine variety, this is equivalent to saying that every regular function vanishing on the subset vanishes on the entire variety.