Let $\mathbb{R}_{Z}^n$ to denote $\mathbb{R}^n$ equipped with the Zariski topology.
(a) Suppose we have parametric line $L$, namely $p(t)=(a_1+tb_1,\dots, a_n+tb_n)$ where $t\in \mathbb{R}$. Prove that the subspace topology that $L$ inherits from $\mathbb{R}_Z^n$ is the finite complement topology.
Sketch: Closed sets in the subspace topology of $L$ has form $L\cap Z(\mathcal{B})$ where $$Z(\mathcal{B})=\{(x_1,\dots, x_n)\in \mathbb{R}^n: p(x_1,\dots,x_n)=0 \ \text{for all} \ p\in \mathcal{B}\}.$$
We need to show that $L\cap Z(\mathcal{B})$ is either finite or L.
If it's finite then OK, suppose $L\cap Z(\mathcal{B})$ is infinite. Then how to prove that it all $L$?
These closed subsets are the zeroes of the restriction of a finite set of polynomial functions $p_1,...,p_m$ to $L$, each of the restriction of $p_i$ to $L$ is a polynomial function of $t$, thus it has a finite set of zeroes or its set of zeroes is $L$. We deduce that such a closed subset is infinite if and only if the restriction of every $p_i$ has an infinite set of zeroes, which implies that the closed subset is $L$.