- #1

Silviu

- 624

- 11

## Homework Statement

Consider the set of real number with the following metric: ##\frac{|x-y|}{1+|x-y|}##. Which subsets of R with this metric are open, closed, bounded or compact?

## Homework Equations

## The Attempt at a Solution

First I calculated the neighborhood in this metric. If the radius of the neighborhood is bigger than 1, it contains the whole R (this implies that all sets are bounded). If a neighborhood has a radius smaller than 1, that it is an interval of the form ##(a-\frac{r}{1-r},a-\frac{r}{1+r})##. Now looking at open sets, a set is open if for every element there is a neighborhood of that element completely in the set and this allows us to take r as small as we want, so even smaller than 1. So if open sets are made of open intervals (open intervals in R with the normal metric), and the union of open intervals is an open interval, it means that open sets are open intervals. For closed sets and compact, we can use the same reasoning as in R with the normal metric as we can take r < 1 and make the substitution ##r \to \frac{r}{1-r}##. So closed and compact sets are the same as in R. Is my reasoning good enough, like is it rigorous, or there are some spaces that need to be filled? Thank you!