Let $k$ be a field, let $n\in \mathbb N$ with $n\ge 1$, and identify affine $n$-space $\mathbb A_k^n$ with the set $k^n$. Let $X\subseteq \mathbb A_k^n$ be any subset. Define its vanishing ideal by $I(X):=\{f\in k[x_1,\dots,x_n]: f(a)=0 \text{ for every } a\in X\}$. Let $k[X]:=k[x_1,\dots,x_n]/I(X)$ be the coordinate ring of polynomial functions on $X$, regarded as a $k$-algebra through the image of the constant polynomials. Then $k[X]$ is reduced and finitely generated as a $k$-algebra.