# Finite subset of a metric space

1. Feb 24, 2008

### rjw5002

1. The problem statement, all variables and given/known data

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

2. Relevant equations

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

3. 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, $$\forall$$n$$\in$$N. Therefore, for any point p, there exists at least one neighborhood that contains no points q $$\in$$ E.

I feel that this is true, but have I made any incorrect assumptions??
Thanks a lot for any feedback.

2. Feb 24, 2008

### morphism

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. Feb 24, 2008

### HallsofIvy

Staff Emeritus
remember that you have the additional rquirement that q$\ne$ 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. Feb 25, 2008

### rjw5002

Right, ok. I can fix that detail. Thanks a lot guys.

5. Sep 1, 2011

### nehakapoor

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. Sep 1, 2011

### micromass

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