Given a field $K$, if we consider affine $n$ space over $K$, denoted $\mathbb{A}^n_K$. Now lets consider $K[z_1,z_2,...,z_n]$, given some ideal $I\subset K[z_1,z_2,...,z_n]$, we associate the vanishing set of $I$ by $$ V(I)=\{(z_1,z_2,...,z_n)\in\mathbb{A}_K^n: f(z_1,z_2,...,z_n)=0\text{, for all $f\in I$}\} $$ Now we can put the Zariski topology on $\mathbb{A}_K^n$ by saying it is the topology whose closed sets are generated by the vanishing sets of ideals $V(I)$ for all ideals $I\subset K[z_1,z_2,...,z_n]$. Put more simply, a closed set is just the simultaneous vanishing of a set of polynomials.
Now my question is we can put another topology on $\mathbb{A}_K^n$ where we restrict to the case where we allow the degree of the polynomial to be at most $1$. That is the closed sets are generated by linear subspaces. I didn't know if this topology had a name, or was useful in any way. Since it is defined in a similar way and is just a coarser topology when compared to the Zariski topology.
