1

Let $(X,d)$ be a metric space. Which are some of the propertis of the metric $$\rho(x_1,x_2)=\dfrac{d(x_1,x_2)}{1+d(x_1,x_2)}$$

I mean, if $(X,d)$ is complete or separable, then one can conclude that $(X,\rho)$ is so? and there is another properties?

Guadalupe
  • 103
  • 2
    $\rho$ has the same Cauchy sequences, the same convergent sequences, and the same limits as $d$. Except it's bounded, whereas $d$ might not be. Perhaps a bit more can be said? – Ian Sep 30 '15 at 02:20
  • That must means that if $(X,d)$ is complete, then $(X,\rho)$ does so, but if every Cauchy sequence is bounded then $(X,\rho)$ would be always complete – Guadalupe Sep 30 '15 at 02:32
  • Indeed: if $d$ is complete then so is $\rho$, and if $d$ is separable then so is $\rho$. – Ian Sep 30 '15 at 02:54
  • And what about the conversely? – Guadalupe Sep 30 '15 at 02:58
  • Sure, the converse holds. It's not that hard to see: first, if $\delta>0$ then $d \leq \delta$ implies $\rho \leq \delta$. On the other hand, if $0<\delta<1$ then $\rho \leq \delta/2$ implies $d \leq \delta$. Thus at least for small distances, the two metrics are essentially the same. They only differ for large distances, which are not relevant to Cauchiness, convergence, separability, etc. – Ian Sep 30 '15 at 03:15

1 Answers1

1

As Ian said,

for small distances, the two metrics are essentially the same. They only differ for large distances, which are not relevant to Cauchiness, convergence, separability, etc.

It may be instructive to compare $\rho$ to the metric $\tilde \rho(x_1,x_2)=\min(1,d(x_1,x_2))$. This definition makes it clear that $\tilde \rho$ is identical to $d$ at small scales. And the metrics $\rho$ and $\tilde \rho$ are comparable, since $ \frac12 \tilde\rho \le \rho \le \tilde \rho $ (which is easy to check).