In their book "Algebraic Geometry I" Gortz and Wedhorn at page 16, after defining the category of affine algebraic sets (Zariski topology in $\mathbb A^n_k$, corrispondence between radical ideals and closed subsets, morphisms of affine algebraic sets ecc...), say:
The notion of an affine algebraic set is still not satisfactory. We list three problems:
Open subsets of affine algebraic sets do not carry the structure of an affine algebraic set in a natural way. In particular we cannot glue affine algebraic sets along open subsets (although this is a “natural operation” for geometric object
Intersections of affine algebraic sets in $\mathbb A^n_k$ are closed and hence again affine algebraic sets. But we cannot distinguish between $V (X) \cap V (Y ) ⊂ \mathbb A^2_k$ and $V (Y ) \cap V (X^2 −Y ) \subset \mathbb A^2_k$ although the geometric situation seems to be different (we will see similar phenomena later when we study fibers of morphisms)
Affine algebraic sets seem not to help in studying solutions of polynomial equations in more general rings than algebraically closed fields.
The first problem is due to the fact that affine algebraic sets are necessarily embedded in an affine space.
Now, for me the third problem is obviously clear. But what is the meaning of the first two problems? I need some further explanations. Moreover, why they say that the problem 1. is due to the embedding in $\mathbb A^n_k$?