Let $d : \mathbb{R}^n \times \mathbb{R}^n \to [0, \infty]$ be defined by $ d(x,y) = \left\{ \begin{array}{ll} 0 & : ~ x = y \\ ||x|| + ||y|| & : ~ x \ne y \end{array} \right. $ where $||\cdot ||$ denotes the usual norm of $\mathbb{R}^n$.
Show that $d$ is a metric.
Draw the $\varepsilon$-Spheres $B_{\varepsilon}(x_0) := \{ x \in \mathbb{R}^2 ~|~ d(x,x_0) < \varepsilon \}$ for $x_0 = (0,0)$ and $x_0 = (1,1)$ and $\varepsilon = \frac{1}{2}, 1, \frac{3}{2}$.
Characterize the open, closed and compact sets with respect to this metric.
Is $(\mathbb{R}^n, d)$ complete?
Number 1) is simple, for 2) I got:
If $x_0 = (0,0)$ then $ d(x,x_0)= \left\{ \begin{array}{ll} 0 & \textrm{ for } x = (0,0) \\ \sqrt{x^2 + y^2} & \textrm{ otherwise } \end{array} \right. $ and if $x_0 = (1,1)$ then $ d(x,x_0) = \left\{ \begin{array}{ll} 0 & \textrm{ for } x = (1,1) \\ \sqrt{2} + \sqrt{x^2 + y^2} & \textrm{ otherwise } \end{array} \right. $ and the pictures are simple spheres with the point $x_0$ in the sphere ($x_0 = (0,0)$) or isolated outside ($x_0 = (1,1)$).
But with 3) I have my problems, i conjecture that $ B_{\varepsilon}(x) \quad \textrm{ is open iff } \quad ||x|| - \varepsilon > 0 $ and going on I know that finite intersections of open sets are open, but then had I got all open sets by this construction? And what about the other properties, how can I characterize them, do you have any hints?