thin algebraic set
Definition 1.
Let $V$ be an irreducible algebraic variety (we assume it to be integral and quasiprojective) over a field $K$ with characteristic zero. We regard $V$ as a topological space^{} with the usual Zariski topology^{}.

1.
A subset $A\subset V(K)$ is said to be of type ${C}_{1}$ if there is a closed subset $W\subset V$, with $W\ne V$, such that $A\subset W(K)$. In other words, $A$ is not dense in $V$ (with respect to the Zariski topology).

2.
A subset $A\subset V(K)$ is said to be of type ${C}_{2}$ if there is an irreducible variety ${V}^{\prime}$ of the same dimension^{} as $V$, and a (generically) surjective^{} algebraic morphism $\varphi :{V}^{\prime}\to V$ of degree $\ge 2$, with $A\subset \varphi ({V}^{\prime}(K))$
Example.
Let $K$ be a field and let $V(K)=\mathbb{A}(K)={\mathbb{A}}^{1}(K)=K$ be the $1$dimensional affine space. Then, the only Zariskiclosed subsets of $V$ are finite subsets of points. Thus, the only subsets of type ${C}_{1}$ are subsets formed by a finite number of points.
Let ${V}^{\prime}(K)=\mathbb{A}(K)$ be affine space and define:
$$\varphi :{V}^{\prime}\to V$$ 
by $\varphi (k)={k}^{2}$. Then $\mathrm{deg}(\varphi )=2$. Thus, the subset:
$$A=\{{k}^{2}:k\in \mathbb{A}(K)\}$$ 
, i.e. $A$ is the subset of perfect squares^{} in $K$, is a subset of type ${C}_{2}$.
Definition 2.
A subset $A$ of an irreducible variety $V\mathrm{/}K$ is said to be a thin algebraic set (or thin set, or “mince” set) if it is a union of a finite number of subsets of type ${C}_{\mathrm{1}}$ and type ${C}_{\mathrm{2}}$.
References
 1 J.P. Serre, Topics in Galois Theory^{}, Research Notes in Mathematics, Jones and Barlett Publishers, London.
Title  thin algebraic set 

Canonical name  ThinAlgebraicSet 
Date of creation  20130322 15:14:13 
Last modified on  20130322 15:14:13 
Owner  alozano (2414) 
Last modified by  alozano (2414) 
Numerical id  5 
Author  alozano (2414) 
Entry type  Definition 
Classification  msc 12E25 
Synonym  thin set 
Synonym  mince set 