# Open Sets in a Discrete Metric Space .... ....

• MHB
• Math Amateur
In summary, open balls in a discrete metric space are either singleton sets or the entire space. The union of any collection of open sets is open, and since any singleton set is an open ball, any union of singleton sets is also open. This means that every set in a discrete metric space is open and closed.
Math Amateur
Gold Member
MHB
In a discrete metric space open balls are either singleton sets or the whole space ...

Is the situation the same for open sets or can there be sets of two, three ... elements ... ?

If there can be two, three ... elements ... how would we prove that they exist ... ?

Essentially, given the metric or distance function, I am struggling to see how in forming a set of the union of two (or more) singleton sets you can avoid including other elements of the space ...

Peter

Last edited:
As you say, open balls are either singleton sets or the entire space. But the union of any collection of open sets are open. Since any singleton sets are open balls (so open sets) any union of singleton sets is open. But any set is a union of singleton sets! Therefore every set is open in the discrete metric. (And every set is closed.)

HallsofIvy said:
As you say, open balls are either singleton sets or the entire space. But the union of any collection of open sets are open. Since any singleton sets are open balls (so open sets) any union of singleton sets is open. But any set is a union of singleton sets! Therefore every set is open in the discrete metric. (And every set is closed.)
Thanks HallsofIvy ...

Peter

## 1. What is a discrete metric space?

A discrete metric space is a mathematical concept that consists of a set of points or elements with a distance function defined between them. In a discrete metric space, the distance between any two points is either 0 or 1, making it a discrete or "separated" space.

## 2. What are open sets in a discrete metric space?

In a discrete metric space, an open set is a collection of points that do not include their boundary points. In other words, for any point in an open set, there exists a small enough radius where all the points within that radius are also in the set.

## 3. How are open sets different from closed sets in a discrete metric space?

Unlike open sets, closed sets in a discrete metric space include their boundary points. This means that for any point in a closed set, there exists a small enough radius where all the points within that radius are also in the set, including the boundary points.

## 4. What is the significance of open sets in a discrete metric space?

In a discrete metric space, open sets play a crucial role in defining continuity and convergence of sequences. They also help in defining concepts such as compactness and connectedness, which are important in many areas of mathematics and science.

## 5. Can you give an example of an open set in a discrete metric space?

One example of an open set in a discrete metric space is the set of all natural numbers. In this set, any point has a small enough radius where all the points within that radius are also in the set. For instance, the point 5 has a radius of 1, where all the points within that radius (4, 5, and 6) are also in the set of natural numbers.

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