Proving the Discreteness of a Metric Space with Open Closure Property"

In summary, the problem states that given a metric M where the closure of every open set is open, one must prove that M is a discrete metric. The attempt at a solution involves showing that every singleton set in M is open, and using the statement about the closure of open sets to do so. The conversation also mentions the importance of proving that an open set is equal to its own closure and choosing the right open set to show that points are open.
  • #1
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Homework Statement


the problem:
Let M be a metric in which the closure of every open set is open. Prove that M is discrete


The Attempt at a Solution


To show M is discrete, it's enough to show every singleton set in M is open.
For any x in M, assume it's not open,
then there exist a converging sequence in M-{x} converges to x

I want to show such sequence does not exist, but I really don't know how to use the original statement that the closure of open set is open

Thank for help
 
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  • #2
First show that under this hypothesis, an open set is in fact equal to its own closure. Then with the right choice of open set it's not hard to see that points are open.
 

1. What is a metric space?

A metric space is a mathematical concept that represents a set of points, where the distance between any two points is defined. It is used to study the properties of objects and spaces in mathematics, physics, and other sciences.

2. What does it mean for a metric space to have an open closure property?

An open closure property means that for any point in the metric space, there exists a neighborhood (open ball) around that point that is contained entirely within the space. In other words, any point in the space has a set of nearby points that are also in the space.

3. How do you prove the discreteness of a metric space with open closure property?

To prove the discreteness of a metric space with open closure property, one must show that every point in the space has a neighborhood (open ball) that contains only a finite number of other points in the space. This can be done by using the definition of an open set and the open closure property of the space.

4. What is the significance of proving the discreteness of a metric space with open closure property?

Proving the discreteness of a metric space with open closure property is important because it allows us to better understand the structure and properties of the space. It also helps in the study of limit points, convergence, and continuity in the space.

5. Can a metric space have an open closure property but not be discrete?

Yes, it is possible for a metric space to have an open closure property but not be discrete. This can occur when there are points in the space that have a neighborhood containing infinitely many other points. In this case, the space would not be discrete, but still have the open closure property.

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