Algebraic subset
Affine
Let $k$ be an algebraically closed field. Let $A=k[x_1,\dots,x_n]$. For $T\subset A$, and define $$Z(T)={P\in \mathbb{A}^n\mid f(P)=0 \text{ for all } f\in T}.$$
Definition. An algebraic subset $V\subseteq\mathbb{A}^n$ is a set of tuples satisfying some collection of polynomials: $$ V={(a_1,\dots,a_n)\mid f_\alpha(a_1,\dots,a_n)=0, f_\alpha\in K[x_1,\dots,x_n]}. $$ Equivalently, a subset $Y$ of $\mathbb{A}^n$ is an _algebraic set_ if there exists $T\subseteq A$ such that $Y=Z(T)$.
Examples:
- $V=\emptyset$ where $f_\alpha$ is any subset of the nonzero constant polynomials
- $V=\mathbb{A}^n$ where $f_\alpha=0$.
- The only algebraic subsets of $\mathbb{A}^1$ are the empty set, the entire set, or finite collections of points. This is due to polynomials in one variable splitting over $K$.
These determine a topology in which they are precisely the closed sets, the Zariski topology.
Definition. An algebraic subset $V\subseteq\mathbb{A}^n$ is called reducible if $V=V_1\cup V_2$, where each $V_i$ is a proper algebraic subset. Otherwise $V$ is called irreducible. This is an Affine variety.
Although similar to the notion of connectedness in topology, the two are different. Namely there are reducible but connected algebraic subsets.
Theorem. Every algebraic subset can be decomposed into finitely many components.
Quasi-affine
Definition. An open subset of an algebraic subset is a quasi-affine algebraic subset.
Projective
Definition. A subset $Y$ of $\mathbb{P}^n$ ( Projective space) is an algebraic set if there exists a set $T$ of homogeneous elements of $S$ such that $Y=Z(T)$.
See Projective space for details.