Let $k$ be an [algebraically closed field](/page/Algebraically%20Closed%20Field), let $n\geq 0$ be an integer, and let $X\subset \mathbb A_k^n$ be a nonempty affine variety over $k$ with coordinate ring $k[X]$. Define $\dim X:=\dim k[X]$, the Krull dimension of its coordinate ring. Then $\dim X$ is the supremum of all integers $r\geq 0$ for which there exists a strictly decreasing chain of irreducible closed subsets $X_0 \supsetneq X_1 \supsetneq \cdots \supsetneq X_r$ of $X$.