Finite SETS

**frenchguy87** · February 2nd 2010, 11:25 AM

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

**Drexel28** · February 2nd 2010, 12:02 PM

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

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

**frenchguy87** · February 2nd 2010, 12:18 PM

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

**Drexel28** · February 2nd 2010, 12:24 PM

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.

**Plato** · February 2nd 2010, 12:36 PM

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?

**frenchguy87** · February 8th 2010, 05:50 AM

Originally Posted by Drexel28

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

I'm not sure where to go from there...

**HallsofIvy** · February 8th 2010, 07:04 AM

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.

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