Interesting Examples of Metrics I've found some interesting metrics other than the euclidean metric, such as the taxi cab metric and the British rail metric. Are there any other interesting metrics out there? Please give some sort of visualization if possible.
 A: *

*The river metric on $\Bbb R^2: \; d((x,y),(x,y'))=|y-y'|$ but if $x\ne x'$ then $d((x,y),(x',y'))=|x-x'|+|y|+|y'|.$ The idea is that if $x\ne x'$ then $\{x\}\times \Bbb R$ is separated from $\{x'\}\times \Bbb R$ by mountains, so to travel from $(x,y)$ to $(x',y')$ you must  travel to , and along the river $\Bbb R\times \{0\}.$

*A metric on $\Bbb R\cup \{p\}$ with $p\not \in \Bbb R:\; d(x,x')=\frac {|x-x'|}{1+|x-x'|}$ when $x,x'\in \Bbb R,$ and $d(p,x)=2-d(0,x)$ when $x\in \Bbb R.$ This gives a locally-compact complete metric space with a closed subset $\Bbb R$  such that $\inf \{d(p,x):x\in \Bbb R\}=1,$ but no $x\in \Bbb R$ satisfies $d(p,x)=1.$

*For $1\le p\in \Bbb R$ the $l_p$ metric $d_p((x,y),(x',y'))=(|x-x'|^p+|y-y'|^p)^{1/p}$ generates the standard topology (i.e. the topology from $p=2$) on $\Bbb R^2.$ As $p\to \infty,$ the unit sphere $\{(x,y): d_p((x,y),(0,0))=1\}$ approaches $\{(x,y):\max (|x|,|y|)=1\},$ which is the unit sphere of the $l_{\infty}$ metric $d_{\infty}((x,y),(x',y'))=\max (|x-x'|,|y-y'|),$ which also generates the same topology. This generalizes to $\Bbb R^n$ for $2\le n \in \Bbb N.$

*If $(X,d)$ and $(Y,e)$ are metric spaces and if $f:X\to Y$ is continuous, then  on $X,$ the metric $d'(x,x')=d(x,x')+e(f(x),f(x'))$ on $X$ is equivalent to $d.$
That is, $d$ and $d'$ generate the same topology on $X$.
Particularly when $Y=\Bbb R$ and $e(y,y')=|y-y'|,$ this is useful.
For example if $(X,d)$ is a non-compact metric space, we show that $X$ has an infinite closed discrete subspace $S$ and show there is a continuous $f:X\to \Bbb R$ such that $\{f(x):x\in S\}$ is unbounded in $\Bbb R,$ in order to prove that the $d$-topology on $X$ can be generated by an unbounded metric, namely by $d'(x,x')=d(x,x')+|f(x)-f(x')|.$

*For any metric space $(X,d)$ the metrics $d_1(x,x')=\min (1,d(x,x'))$ and $d_2(x,x')=\frac {d(x,x')}{1+d(x,x')}$ are equivalent to $d.$ This illustrates that "bounded subset of $X$" is not generally a topological property, but (unless $X$ is compact) depends on the metric, not on the topology that the metric generates.
