Answer to Question #217017 in Differential Geometry | Topology for Prathibha Rose

Question

Give examples of two metrics on R ^2 which are not equivalent .substantiate your claim

Answer 1

Solution:

The following metrics on $\mathbb{R}^{2}$ are not equivalent to one another: the Euclidean metric d, the hub metric $\varrho_{h}$ , and the discrete metric $\varrho_{\text {disc }}$. These are defined as follows:

The euclidean metric on $\mathbb{R}^{2}$ is defined by

$d(\underline{x}, \underline{y})=\sqrt{\left(x_{1}-y_{1}\right)^{2}+\left(x_{2}-y_{2}\right)^{2}},$

where $\underline{x}=\left(x_{1}, x_{2}\right)$ and $\underline{y}=\left(y_{1}, y_{2}\right)$ . 

Let $X=\mathbb{R}^{2}$ . For $x=\left(x_{1}, x_{2}\right), y=\left(y_{1},y_{2}\right)$ define $\varrho_{h}(x, y)$ as follows. If x=y then $\varrho_{h}(x, y)=0$ . If $x\ne y$ then

$\varrho_{h}(x, y)=\sqrt{x_{1}^{2}+x_{2}^{2}}+\sqrt{y_{1}^{2}+y_{2}^{2}}$

The metric $\varrho_{h}$ is called the hub metric on $\mathbb{R}^{2}$ .

The discrete metric on an arbitrary set X is defined by

$d(x, y)=\left\{\begin{array}{lll} 0 & \text { if } & x=y \\ 1 & \text { if } & x \neq y \end{array}\right.$

Every metric defines open balls, but even if metrics are equivalent their open balls may look very differently (compare e.g. open balls in $\mathbb{R}^{2}$ taken with respect to d and $\left.\varrho_{\text {ort }}\right)$ . It turns out, however, that each metric defines also a collection of so-called open sets, and that open sets defined by two metrics are the same precisely when these metrics are equivalent.