Difference of affine varities $\mathbb V(xz,yz)-\mathbb V(z)$ not variety? Consider the ideal $V_1:=\mathbb V(xz,yz)$, xy -plane union z -axis, and $\mathbb V(z)$, z -axis. Now the difference $\mathbb V_1-\mathbb V(z)$ equals to $z$ -axis with origin moved. The z-axis is the smalles variety containing $V_{not}:=\mathbb V_2-\mathbb V(z)$. Variety is defined such that 

Let $k$ be a field, and let $f_1,\ldots ,f_s$ be polynomials in $k[x_1,\ldots,x_n]$. Then 
  $$\mathbb V(f_1,\ldots,f_s) = \{(a_1,\ldots,a_n)\in k^n : f_i(a_1,\ldots,a_n) = 0 \forall 1\leq i\leq s\}.$$ We call $V(f_1,\ldots,f_s)$ the affine variety defined by $f_1,\ldots,f_s$.

Please explain why this $V_{not}$ is not a variety.
P.s. Studying the book Ideals, Varities, and Algorithms by Cox et all (2008, 3rd edition): the example given in introduction Ch1 and then this observation related to Zariski closure, elimination and quotients ~Ch3.
 A: If $k$ is a finite field, your set is indeed a variety. If $k$ is infinite,
let $\{(0,0,z) ~|~ z \neq 0\} \subset V(f_1, \dotsc, f_s)$, i.e. $f_i(0,0,z)=0$ for any  $z \in k\setminus \{0\}$. Since $k$ is infinite, this shows that $f_i(0,0,z) \in k[z]$ is the zero-polynomial, i.e. we have $f(0,0,0)=0$. This shows that $(0,0,0)$ is a point of $V(f_1, \dotsc, f_s)$, hence $\{(0,0,z) ~|~ z \neq 0\}$ is not an affine variety.
If the field happens to be $\mathbb R$ or $\mathbb C$, we can shorten the argument and just notice that any affine variety is closed with respect to the euclidean topology.
A: It is not a (closed) variety simply because it is not the zero locus of a collection of polynomials. Given two varieties $V(f_1,\dots,f_m),V(g_1,\dots,g_n)$, the union and intersection is a variety because the union is the zero locus of the products $V(f_i g_j)$ and the intersection is the zero locus of both sets of polynomials $V(f_1,\dots,f_m, g_1, \dots g_n)$. We cannot in general define the set difference as a zero locus. $V_{\text{not}}$ is however an open subset of a variety, so it is a quasiaffine variety as they are called.
