Classical affine algebraic sets, including the empty boundaries

Fix an algebraically closed field $k$ and $n\ge 0$. Write $R=k[x_1,\dots ,x_n]$ for the iterated polynomial ring of Polynomial rings in finitely many commuting indeterminates by iteration. Algebraic closure has the meaning of An algebraically closed field: every nonconstant polynomial has a root in the field. For any $S\subseteq R$, define $V(S)=\{a\in k^n:\text{for every }f\in S,f(a)=0\}\mathrm{.}$ An affine algebraic set is any such subset, including the empty set. Here $k^0=\{()\}$ and $R=k$ when $n=0$. With the abbreviations $V(0):=V(\{0\})$ and $V(1):=V(\{1\})$, one has $V(\mathrm{\varnothing })=V(0)=k^n$ and $V(1)=\mathrm{\varnothing }$: an empty list of equations imposes no condition, whereas $1(a)=1\mathrm{\ne }0$. Evaluation, as in Evaluation and roots of a polynomial in a commutative target ring, is performed successively in the finitely many variables.

Source comparison: Milne, Algebraic Geometry, v6.10, §2a, pp. 36–37. Conventions here distinguish arbitrary affine algebraic sets from nonempty irreducible varieties.