Proposition

All finite subsets of $k^n$ are affine varieties.

1. Prove that a single point $(a_1, \cdots, a_n) \in k^n$ is an affine variety.
2. Prove that every finite subset of $k^n$ is an affine variety.

Solution

1

Let $(a_1, \cdots, a_n) \in k^n$ be given. Let $S = V(x_1 - a_1, \cdots, x_n - a_n)$. Then $S = { (a_1, \cdots, a_n) }$. Thus every singleton is an affine variety.

2

Lemma 2 (P.11, Ideals, Varieties and Algorithms) states that the union of two affine varieties is an affine variety.

By mathematical induction, for any $m \in \mathbb{N}$, a finite subset of $k^n$ with $m$ points is an affine variety.