# Exploring Open, Closed, Bounded, and Compact Sets in R with a Unique Metric

• Silviu
In summary, the conversation discusses the set of real numbers with a given metric and explores which subsets of this set are open, closed, bounded, and compact. The proof hinges upon calculations involving the metrics and the definitions of open, closed, and compact sets. The reasoning presented is considered sound, but could benefit from additional polishing and rigor.
Silviu

## 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?

## 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!

It could use some polishing, but I think you are heading in the right direction. I suppose the rigor of your proof hinges upon these calculations you performed but didn't include. Let ##g(x,y) = \frac{|x-y|}{1+|x-y|}## and let ##d(x,y) = |x-y|##, which is the standard metric. For boundedness, you have to be careful. A subset nonempty subset ##S## of real numbers is bounded provided there exists some ##S > 0## such that ##g(x,y) \le M## for all ##x,y \in S##. So I don't think you actually proved that all sets are bounded, although you are right that all sets are bounded (in fact, ##g(x,y)## is typically called the standard bounded metric).

Regarding the closed and compact sets, there isn't really anymore work to do once you have shown that open sets generated by the (bounded) metric ##g## are the same as those in the generated by the standard (Euclidean) metric ##d##. To do this, given

$$B_g(x,\epsilon) := \{y \in \Bbb{R} \mid g(x,y) < \epsilon \} = \{ y \in \Bbb{R} \mid \frac{|x-y|}{1 + |x-y|} < \epsilon \}$$ (i.e., the ##\epsilon## ball with respect to the metric ##g##), you need to find a ##\delta > 0## such that

$$B_{d}(x,\delta) \subseteq B_g(x,\epsilon),$$

where ##B_d(x,\delta) = \{ y \in \Bbb{R} \mid d(x,y) < \delta \} = \{y \in \Bbb{R} \mid |x-y| < \delta \}##. Then you have to do the samething with the roles of ##g## and ##d## reversed (make sure you draw lots of pictures--they will help you find the ##\delta > 0## and ##\epsilon > 0## you need).

Perhaps you ought to include a precise formulation of your definitions of open, closed, and compact sets.

## 1. What is a metric on R?

A metric on R is a function that assigns a non-negative real number to pairs of real numbers, satisfying three properties: non-negativity, symmetry, and the triangle inequality. Essentially, it is a way to measure the distance between two points on the real number line.

## 2. What are the different types of metrics on R?

There are several types of metrics on R, including the Euclidean metric, the taxicab metric, the Chebyshev metric, and the Minkowski metric. Each of these metrics has different ways of measuring distance and is useful for different applications.

## 3. How do metrics on R differ from each other?

Metrics on R differ in the way they measure distance between points. For example, the Euclidean metric is the most commonly used and measures the straight-line distance between two points, while the Minkowski metric is used in relativity and measures the spacetime interval between two events.

## 4. Can I use any metric on R?

Yes, you can use any metric on R as long as it satisfies the three properties mentioned earlier: non-negativity, symmetry, and the triangle inequality. However, some metrics may be more appropriate for certain situations, so it is important to choose the right metric for your specific application.

## 5. How are metrics on R used in science?

Metrics on R are used in various scientific fields, such as physics, engineering, and mathematics. They are used to measure distances, define topology, and study relationships between objects in space. They also have practical applications in data analysis, optimization, and machine learning.

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