Let $\tau_1$ be the topology on $\mathbb{R}^2$ induced by the Euclidean norm.
We consoder the seminorms $\{s_k;\ k=1,2\}$ on $\mathbb{R}^2$ defined by $$ s_1(x)=|x_1-x_2|,\quad s_2(x)=|x_1+x_2|,\quad\forall x=(x_1,x_2)\in \mathbb{R}^2 $$ We have a neighborhood $U$ of $(0,0)$ is $$ U=\cap_{j\in\{1,2\}}\{x;\ s_j(x)\le \epsilon_j \} $$ where $\epsilon_j >0$. Then, a neighborhood of $x$ is $x+U$, $x\in\mathbb{R}^2$.
We define a subset $G\subset \mathbb{R}^2$ an open set if $G$ contains a neighborhood of each of its point.
It is easy to show that the collection $\tau_2$ of all open sets defined above forms a topology on $\mathbb{R}^2$ and Hausdorff space
How to show that the topology $\tau_1$ is identical with $\tau_2$?