Consider $\mathbb{R}^{2}$ equipped with the discrete metric

$$d(x, y)= \begin{cases}1 & \text { if } x eq y \\ 0 & \text { if } x=y\end{cases}$$

where $x$ and $y$ are elements of $\mathbb{R}^{2}$, and with the corresponding metric topology $\tau_{d}$. Is $\left(\mathbb{R}^{2}, \tau_{d}\right)$ connected? Give a proof. Show also that $d(x, y)$ satisfies the definition of a metric.