Infinite Sets and Cluster Points

**frenchguy87** · February 14th 2010, 07:36 AM

How would you show that an infinite set $\mathbb{N}$ has no cluster points...

**Plato** · February 14th 2010, 07:53 AM

Originally Posted by frenchguy87

How would you show that an infinite set $\mathbb{N}$ has no cluster points...

If $n\in\mathbb{N}$ can the open set $\left(n-\frac{1}{2},n+\frac{1}{2}\right)$ contain an point of $\mathbb{N}$ other than $n$ ?

If $x\notin\mathbb{N}$ is there an open set containing $x$ and no point of $\mathbb{N}$ .
Hint: $\left( {\exists j \in \mathbb{Z}} \right)\left[ {j < x < j + 1} \right]$ .
Let $\delta = \min \left\{ {\left| {x - j} \right|,\left| {j + 1 - x} \right|} \right\}$

**Drexel28** · February 15th 2010, 06:57 PM

Originally Posted by frenchguy87

How would you show that an infinite set $\mathbb{N}$ has no cluster points...

A little more generally, any set $E$ such that $E$ such that $d(e,e')>\delta>0$ for all $e,e'\in E$ has no limit points. To see this let $x$ be an arbitrary point and consider $B_{\frac{\delta}{2}}(x)$ . It either contains one or zero points of $E$ . If the former we're done, if the latter take the open ball of one-half the distance from $x$ to that point.

Thread: Infinite Sets and Cluster Points

LinkBack

Thread Tools

Display

Infinite Sets and Cluster Points

Similar Math Help Forum Discussions

lim sup, lim inf and cluster points.

Quick Cluster Points Question

Cluster Points in a Sequence

Definitions of accumulation points,limit points,cluster points

Need Help with Cluster Points

Search Tags