Counter example of Algebraic sets Affine n - spaces over a field $K$ is the cartesian product of the field $K$ with itself $n$ time and it is denoted by $\mathbb A^n(K)$. $X$ is a subset of Afine n - spaces $\mathbb A^n(K)$ is called algebraic if there exist an ideal I of $K[x_1,X_2, \cdots , X_n]$ such that $V(I) = X$, where $V(I) = \\{f \in I : f(a_1, a_2 , \cdots, a_n) = 0 \ \ \text{for all} \ (a_1, \cdots , a_n) \in X \\}$ > > Give an example of countable collection of algebraic sets whose union is not algebriac.

Let $K=\mathbb R$ and let $a_i\in\mathbb R$ be infinitely many distinct points indexed by positive integers. Then the union of these points is not algebraic in $A^1$ because every ideal in the polynomial ring is principal and hence every algebraic set is finite.