Finite subset of a metric space

In summary, the conversation discusses the concept of limit points in an infinite set X and how to find them in a finite subset E. The defined equation for d(p,q) and the requirement for a point to be a limit point of E are provided. The attempt at a solution suggests that there are no limit points in any subset of X, but further clarification is given by pointing out that B(p, 1/n) may not be empty for all n. The topic of proving that every finite set is a closed set and compact in a metric space is mentioned, but it is advised to create a new topic for this and provide an attempt at the proof.
  • #1
rjw5002

Homework Statement



Let X be an infinite set. For p,q [tex]\in[/tex] X define:
d(p,q) = {1 if p [tex]\neq[/tex] q; 0 if p = q
Suppose E is a finite subset of X, find all limit points of E.

Homework Equations



definition: a point p is a limit point of E if every neighborhood of p contains a point q [tex]\neq[/tex] p s.t. q[tex]\in[/tex]E

The Attempt at a Solution


My thoughts were that there are no limit points for any subset of X because for any point p, B(p, 1/n) is empty, [tex]\forall[/tex]n[tex]\in[/tex]N. Therefore, for any point p, there exists at least one neighborhood that contains no points q [tex]\in[/tex] E.

I feel that this is true, but have I made any incorrect assumptions??
Thanks a lot for any feedback.
 
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  • #2
B(p,1/n) isn't empty for all n. For instance p is always in B(p,1/n), and if n=1, then B(p,1/n) is in fact all of X.

You definitely have the right idea, but you have to be a bit more careful.
 
  • #3
remember that you have the additional rquirement that q[itex]\ne[/itex] p. I think that is what you meant to say. For all n, B(p, 1/n) contains p, but for some n, B(p, 1/n) does not include any other point of the set.
 
  • #4
Right, ok. I can fix that detail. Thanks a lot guys.
 
  • #5
i need the proof for every finite set is a closed set in a metric space ..
and finite set in a metric space in compact as soon as possible../.
 
  • #6
nehakapoor said:
i need the proof for every finite set is a closed set in a metric space ..
and finite set in a metric space in compact as soon as possible../.

You will need to make a new topic for this. And it would also be nice to see your attempt...
 

1. What is a finite subset of a metric space?

A finite subset of a metric space is a set of elements that is both finite in size and satisfies the properties of a metric space. This means that the set contains a finite number of points and the distance between any two points in the set is well-defined.

2. How is a finite subset of a metric space different from a general metric space?

A finite subset of a metric space differs from a general metric space in that it has a limited number of points, while a general metric space can have an infinite number of points. Additionally, a finite subset may not contain all the properties of a general metric space, such as completeness or connectedness.

3. What is an example of a finite subset of a metric space?

An example of a finite subset of a metric space is a set of points on a line segment. For instance, a set of 5 points located at equally spaced intervals on a line segment would be a finite subset of a metric space.

4. Can a finite subset of a metric space form a metric space on its own?

No, a finite subset of a metric space cannot form a metric space on its own. It must be a part of a larger metric space in order to satisfy all the properties of a metric space, such as the triangle inequality.

5. How are finite subsets of a metric space used in real-world applications?

Finite subsets of a metric space are commonly used in data analysis and machine learning algorithms. They can represent a finite set of data points or observations, and the properties of a metric space can be used to calculate distances and similarities between these data points.

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