(i) Let $t>0$. For $\mathbf{x}=(x, y), \mathbf{x}^{\prime}=\left(x^{\prime}, y^{\prime}\right) \in \mathbb{R}^{2}$, let

$$\begin{gathered} d\left(\mathbf{x}, \mathbf{x}^{\prime}\right)=\left|x^{\prime}-x\right|+t\left|y^{\prime}-y\right|, \\ \delta\left(\mathbf{x}, \mathbf{x}^{\prime}\right)=\sqrt{\left(x^{\prime}-x\right)^{2}+\left(y^{\prime}-y\right)^{2}} \end{gathered}$$

( $\delta$ is the usual Euclidean metric on $\mathbb{R}^{2}$.) Show that $d$ is a metric on $\mathbb{R}^{2}$ and that the two metrics $d, \delta$ give rise to the same topology on $\mathbb{R}^{2}$.

(ii) Give an example of a topology on $\mathbb{R}^{2}$, different from the one in (i), whose induced topology (subspace topology) on the $x$-axis is the usual topology (the one defined by the metric $\left.d\left(x, x^{\prime}\right)=\left|x^{\prime}-x\right|\right)$. Justify your answer.