# Complete/Connected space

The metric $$d: \mathbb{R}^2 \times \mathbb{R}^2 \rightarrow \mathbb{R}$$ defined by

$$d(x, y) = \left\{ \begin{array}{lr} ||x||+||y|| & \mbox{if } x \neq y \\ 0 & \mbox{if } x = y \\ \end{array} \right.$$ is a variation of the SNCF metric. Given that $$d$$ is a metric, show that if $$X = \mathbb{R}^2$$, then $$(X, d)$$ is a complete but not connected metric space.

My attempt:

Complete:

A Metric Space is said to be complete if every Cauchy Sequence converges in the space.

Let $$\{x_n\}$$ be a Cauchy sequence in $$d$$. We need to show that for all $$\epsilon > 0$$, there is an $$N \in \mathbb{N}$$ so that $$d(x_n, x) < \epsilon$$ (for some $$x \in X$$) for all $$n \geq N$$.

Not very sure how to proceed here.

Not connected:

$$(X, d)$$ is disconnected if it can be written as the union of two non-empty separated sets. We need to find $$\emptyset \neq A, \emptyset \neq B \subseteq X$$ so that $$X = A \cup B$$ where $$\overline{A} \cap B = A \cap \overline{B} = \emptyset$$.

I cannot find any such subsets here.

Any assistance is much appreciated.

The most straightforward approach is to figure out what topology on $$\Bbb R^2$$ is generated by $$d$$.
HINT: If $$x\ne\langle 0,0\rangle$$, and $$0, what is $$B(x,r)$$? And what is $$B(\langle 0,0\rangle,r)$$? You should try to figure this out on your own, but I left a description, without proof, in a spoiler box in my answer to your previous question about this metric.
HINT: Every Cauchy sequence in $$\left\langle\Bbb R^2,d\right\rangle$$ either converges to the origin or is eventually constant.