Metrics

due by Wednesday, Sep 3, 2025

We define the SNCF metric $d_{SNCF}$ on $\R^2$. We use polar coordinates on $\R^2$, with $O$ as the origin and coordinates $(r, \theta)$ with $r > 0$ and $\theta\in [0, 2\pi)$ for points in $\R^2\setminus\{O\}$. The metric is defined by:

• $\circ$ $d_{SNCF}(O,O) =0$.
• $\circ$ $d_{SNCF}(O, (r, \theta)) = d_{SNCF}((r, \theta), O) = r$.
• $\circ$ $d_{SNCF}((r_1, \theta_1), (r_2, \theta_2)) = \begin{cases} |r_2 - r_1|,\ \textrm{if $\theta_1 = \theta_2$,} \\ r_2 + r_1,\ \textrm{ if $\theta_1 \neq \theta_2$,} \end{cases}$

Let $d_E$ denote the usual Euclidean metric on $\R^2$.

1. Prove that $d_{SNCF}$ is a metric.
2. Prove or disprove: If a set $U\subset \R^2$ is open in the topology corresponding to $d_{SNCF}$, then $U$ is open in the topology corresponding to $d_E$.
3. Prove or disprove: If a set $U\subset \R^2$ is open in the topology corresponding to $d_E$, then $U$ is open in the topology corresponding to $d_{SNCF}$.