Zero-dimensional varieties are finite sets

Given: The objects and hypotheses in the statement.

For a classical variety $X$, let $\dim X$ be its chain dimension. If $X_1,\dots ,X_m$ are its irreducible components and $x\in X$ is a closed point, define ${\dim }_{x}X={\max }_{x\in X_i}\dim X_i$. The indexing family is nonempty. Say that $X$ has pure dimension $d$ if every irreducible component has dimension $d$; the condition on components is vacuous for the empty variety, whose dimension is nevertheless $-\mathrm{\infty }$. Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Global and local dimension of classical varieties).

Under the Axiom of Choice (The Axiom of Choice), every classical variety is Noetherian and has finitely many irreducible components (Classical varieties have finite irreducible decompositions). This is the only inherited use of Choice in the proof; choosing a point from each member of a finite component family needs no additional choice principle.