If the topology is locally compact and non-trivial, then the metric is usually translation-invariant or even arises from a norm. But is it necessary to be invariant? I failed to give a counterexample.
3
-
4$\begingroup$ On $\mathbb{R}$, you can take e.g. $d(x,y):=|f(x) - f(y)|$ for any monotone homeomorphism $f:\mathbb{R}\to\mathbb{R}$. $\endgroup$– Tobias FritzCommented Sep 24, 2017 at 15:30
-
1$\begingroup$ Take $\mathbb R$ with any metric quasi-isometric to the standard one. $\endgroup$– R WCommented Sep 24, 2017 at 15:30
-
1$\begingroup$ It may be worth noting that one can also construct non-standard uniform structures on $\mathbb{R}$ that induce the standard topology, e.g. by using my construction with $f$ bounded. $\endgroup$– Tobias FritzCommented Sep 24, 2017 at 16:11
Add a comment
|