Let $k$ be a field, let $n\geq 0$ be an integer, set $R:=k[x_1,\dots,x_n]$, and write $\mathbb A_k^n:=k^n$. For a subset $E\subseteq R$, define its zero set by $V(E):=\{a\in \mathbb A_k^n:f(a)=0\text{ for every }f\in E\}$. Equip $\mathbb A_k^n$ with the Zariski topology whose closed subsets are precisely the sets $V(E)$ for $E\subseteq R$, equivalently the sets $V(J)$ for ideals $J\trianglelefteq R$. Then $\mathbb A_k^n$ is a Noetherian [topological space](/page/Topological%20Space): every descending chain $F_1\supseteq F_2\supseteq F_3\supseteq \cdots$ of Zariski closed subsets of $\mathbb A_k^n$ stabilizes.