The Zariski topology is a family of topologies in which algebraic equations, rather than metric inequalities, determine closed sets. It is the basic language for treating solution sets of polynomial equations as geometric spaces. The affine version lives on $\mathbb A^n_k$, the spectrum version lives on $\operatorname{Spec}R$, the projective version lives on projective space using homogeneous equations, and varieties or schemes inherit the topology from these local models. These settings are related, but they are not the same ambient definition. What they share is the governing principle that closed subsets are the loci where algebraic functions vanish.

The guiding principle is that an algebraic condition should define a closed locus. For example, the equation $x^2+y^2-1=0$ cuts out a curve in the affine plane. Several equations cut out the set where all of them vanish at the same time. The Zariski topology turns these common zero sets into the closed subsets of [affine space](/page/Affine%20Space). This viewpoint links topology with ideals, because a system of polynomial equations is best organized by the ideal it generates.

## Definition

In classical algebraic geometry, the unqualified phrase "the Zariski topology" most often means the topology on an affine algebraic set. Spectrum and projective versions use the same guiding principle, but they have their own ambient definitions: spectra use containment in prime ideals, and projective space uses homogeneous equations. We start with the affine definition because it is the canonical model from which the later variants are compared.

[definition: Zariski Topology] Let $k$ be a field, let $X\subset\mathbb A^n_k$ be an affine algebraic set, and let $k[x_1,\ldots,x_n]$ be the coordinate [polynomial ring](/page/Polynomial%20Ring) of the ambient affine space. The Zariski topology on $X$ is the topology whose closed subsets are precisely the sets \begin{align*} X\cap V(I)=\{a\in X:f(a)=0\text{ for every }f\in I\}, \end{align*} where $I$ ranges over ideals of $k[x_1,\ldots,x_n]$. [/definition]

When $X=\mathbb A^n_k$, this says that the closed subsets are exactly the common zero loci $V(I)$. The spectrum and projective versions later in the page keep the same equation-controlled idea, but they change the ambient points and the kind of equations being used.

The ambient affine case is the test model for every later version. When $X=\mathbb A^n_k$, the definition specializes to this concrete rule: the closed subsets are the sets

\begin{align*} V(I)=\{a\in\mathbb A^n_k:f(a)=0\text{ for every }f\in I\}, \end{align*}

where $I$ ranges over ideals of $k[x_1,\ldots,x_n]$. This definition is compact, but it hides the two ingredients that drive the rest of the page: how zero loci behave under operations on ideals, and how much information is lost when equations are replaced by their solution sets. We now unpack those ingredients from the affine case outward.

## Affine Algebraic Sets

We begin over a field $k$, usually algebraically closed when geometric intuition is needed. Affine $n$-space over $k$, written $\mathbb A^n_k$, is the set $k^n$ together with its polynomial coordinate functions. The [coordinate ring](/page/Coordinate%20Ring) is $k[x_1,\ldots,x_n]$. A point $a=(a_1,\ldots,a_n)$ evaluates a polynomial $f$ by substitution:

\begin{align*} f(a)=f(a_1,\ldots,a_n). \end{align*}

The first construction turns a set of equations into its common zero locus. It gives the closed sets before we have named the topology.

[definition: Vanishing Set] The vanishing-set map is \begin{align*} V:\mathcal P(k[x_1,\ldots,x_n])\to \mathcal P(\mathbb A^n_k). \end{align*} For $S\subset k[x_1,\ldots,x_n]$, \begin{align*} S\mapsto V(S)=\{a\in \mathbb A^n_k: f(a)=0\text{ for every }f\in S\}. \end{align*} [/definition]

When $S=\{f\}$, it is standard to write $V(f)$ for $V(\{f\})$. Since $V(\{f\})=V((f))$, the same notation also means the vanishing set of the principal ideal generated by $f$.

[example: A First Vanishing Set] In $\mathbb A^2_k$, write a point as $(a,b)$ with $a,b\in k$. Evaluating the polynomial $y-x^2$ at this point means substituting $x=a$ and $y=b$, so \begin{align*} (y-x^2)(a,b)=b-a^2. \end{align*} By the definition of a vanishing set, $(a,b)\in V(y-x^2)$ exactly when $(y-x^2)(a,b)=0$. Therefore \begin{align*} (a,b)\in V(y-x^2)\quad\Longleftrightarrow\quad b-a^2=0\quad\Longleftrightarrow\quad b=a^2. \end{align*} Thus every point of $V(y-x^2)$ has the form $(a,a^2)$ with $a\in k$. Conversely, for every $t\in k$, \begin{align*} (y-x^2)(t,t^2)=t^2-t^2=0, \end{align*} so $(t,t^2)\in V(y-x^2)$. Hence \begin{align*} V(y-x^2)=\{(t,t^2):t\in k\}. \end{align*} For the polynomial $xy$, substitution gives \begin{align*} (xy)(a,b)=ab. \end{align*} Again by the definition of a vanishing set, \begin{align*} (a,b)\in V(xy)\quad\Longleftrightarrow\quad ab=0. \end{align*} Since $k$ is a field, it has no zero divisors, so $ab=0$ holds exactly when $a=0$ or $b=0$. If $a=0$, then $(a,b)=(0,b)$ lies on the $y$-axis; if $b=0$, then $(a,b)=(a,0)$ lies on the $x$-axis. Therefore \begin{align*} V(xy)=\{(0,b):b\in k\}\cup\{(a,0):a\in k\}. \end{align*} So $V(y-x^2)$ is the parabola parametrized by $t\mapsto(t,t^2)$, while $V(xy)$ is the union of the two coordinate axes. [/example]

This first example shows how single equations and reducible equations appear as closed sets. It also previews two recurring themes: equations may define geometric pieces of different shapes, and factorization in the coordinate ring affects the topology.

The same [closed set](/page/Closed%20Set) is obtained from the ideal generated by $S$. Indeed, if a point kills every polynomial in $S$, it kills every polynomial combination of elements of $S$. For this reason, algebraic geometry usually writes $V(I)$ with $I$ an ideal.

To make these algebraic sets into the closed sets of a topology, two closure properties must be checked. First, unions of finitely many solution sets should again be a solution set; second, arbitrary intersections of solution sets should again be a solution set. The possible obstruction is that union and intersection are operations on sets of points, while the data defining those sets are ideals of polynomials. The key point is that products and sums of ideals translate these set-theoretic operations back into equations.

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