Definition:Affine Algebraic Set

Definition
Let $k$ be a field.

Let $A = k[X_1,\ldots,X_n]$ be the Ring of Polynomial Functions in $n$ variables over $k$.

Then a subset $X \subseteq k^n$ is an affine algebraic set if it is the zero locus of some set $T \subseteq A$.

By Zero Locus of set is Zero Locus of Ideal this is equivalent to saying that there is an ideal $J \subseteq A$ such that $X$ is the zero locus of $J$.