Let $k$ be a field, let $n \in \mathbb{N}$, and set $R := k[x_1, \ldots, x_n]$. For an ideal $I \trianglelefteq R$, define

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

Then the collection of subsets of $\mathbb{A}^n_k$ of the form $V(I)$ is closed under arbitrary intersections and finite unions. More precisely, for ideals $I,J \trianglelefteq R$,

\begin{align*} V(I) \cup V(J) = V(IJ). \end{align*}

For every indexed family of ideals $(I_\lambda)_{\lambda \in \Lambda}$ in $R$,

\begin{align*} \bigcap_{\lambda \in \Lambda} V(I_\lambda) = V\left(\sum_{\lambda \in \Lambda} I_\lambda\right). \end{align*}

Here $IJ$ denotes the product ideal generated by all products $fg$ with $f \in I$ and $g \in J$. The ideal $\sum_{\lambda \in \Lambda} I_\lambda$ consists of finite sums of elements drawn from the ideals $I_\lambda$; when $\Lambda=\varnothing$, this sum is the zero ideal $(0)$, and the corresponding intersection is interpreted as all of $\mathbb{A}^n_k$.