1. ## Finite SETS

How would you show that a finite set has no cluster points?

2. Originally Posted by frenchguy87
How would you show that a finite set has no cluster points?
Is this in a topological space? Or a metric space?

If this is a topological space, this is wrong.

Think about $\displaystyle \left(X,\mathfrak{J}\right)$ where $\displaystyle X=\left\{a,b\right\},\text{ }\mathfrak{J}=\left\{\varnothing,X\right\}$. It is easy to see that $\displaystyle b$ is a limit point of the set $\displaystyle \{a\}$. For, the only neighborhood of $\displaystyle b$ is $\displaystyle X$ which contains a point of $\displaystyle \{a\}$ distinct from itself.

In a metric space $\displaystyle X$ though, think about the set $\displaystyle E=\left\{e_1,\cdots,e_n\right\}$ and for any $\displaystyle x\in X$ take $\displaystyle \delta=\min_{1\leqslant j\leqslant n}d\left(x,e_j\right)$ and consider $\displaystyle B_{\delta}(x)$

3. It doesn't tell you in the problem but since I need to show the statement is true, I will assume topological space.

4. Originally Posted by frenchguy87
It doesn't tell you in the problem but since I need to show the statement is true, I will assume metric space.

5. Originally Posted by frenchguy87
It doesn't tell you in the problem but since I need to show the statement is true, I will assume topological space.
In this case you need to tell us the general setting for the question.
What course is this for, its name?
What is the definition of cluster point?

6. Originally Posted by Drexel28
In a metric space $\displaystyle X$ though, think about the set $\displaystyle E=\left\{e_1,\cdots,e_n\right\}$ and for any $\displaystyle x\in X$ take $\displaystyle \delta=\min_{1\leqslant j\leqslant n}d\left(x,e_j\right)$ and consider $\displaystyle B_{\delta}(x)$
I'm not sure where to go from there...

7. x is a cluster point of A if every neighborhood of x contains some point of A other than x. Since A contains only a finite number of points, there can be only a finite number of points of A in any neighborhood of x. That means that, in any neighborhood of x, there is a point of A, other than x itself, which is closest to x. Look at a neighborhood of x with radius less than that least distance.