Call a metric $\displaystyle d $ compatible on$\displaystyle \mathbb{R}^{n} $ with the vector space structure if $\displaystyle d(x+z, y+z) = d(x,y) $ for all $\displaystyle x,y,z \in \mathbb{R}^{n} $ and $\displaystyle d (\lambda x, \lambda y) = \lambda d(x,y) $ for $\displaystyle \lambda \in [0, \infty) $. Prove that any such metric puts the same topology on $\displaystyle \mathbb{R}^{n} $ as the Euclidean metric.