I'm investigating a topology on that rational plane $\mathbb{Q}^2$, with a subbase which I've had a hard time getting my hands on.
Suppose $X=\{(x,y)\in\mathbb{Q}^2\mid y\geq 0\}$ be the given subset, and let's fix some irrational $j$. Also, I equip $X$ with the topology $\mathcal{T}$ generated by the subbase consisting of sets of form $ A_\epsilon(x,y)=\{(x,y)\}\cup\{(q,0)\mid q\in\mathbb{Q},\left|q-\left(x+\frac{y}{j}\right)\right|<\epsilon\}\cup\{(q,0)\mid q\in\mathbb{Q},\left|q-\left(x-\frac{y}{j}\right)\right|<\epsilon\}. $
So I know that a basic set in this topology consists of finite intersections of such sets, and an element $(x,y)$ is in a basic set if it's in at least one of the three types of sets being unioned to form the subbase sets, for each subbase set in the finite intersection.
With this, why is the space $T_2$? The latter two sets used to make the subbase sets are making it difficult for me to find open sets separating two distinct points.