# Homework Help: Cluster point confusion

1. Nov 30, 2005

### happyg1

cluster point confusion....

Fog Fog Fog....
Ok,Here's the question:
Let A denote the open interval (0,1).Show that the set of Cluster points of A in $$R^1$$ is [0,1].
Our textbook sez that $$R^1$$ is defined as the absolute value metric, i.e. $$\rho$$(x,y)=|x-y|
OK
So I know (and have proven) that (0,1) is uncountable and that there are infinitely many points between any 2 points in (0,1). It is easy for me to understand that the definition of a cluster point
we which we have as:
"let M,$$\rho$$ be a metric space and suppose A $$\subset$$M. The point a $$\in$$M is called a cluster point of A in M if, for every h>0, there exists a point x$$\in$$A such that 0,$$\rho$$(x,a)<h."
is fulfilled....no matter what h I pick, I will always be able to find some x....
I just don't know where to start to write it down in a fashion that my prof. would accept. He's VERY picky and I'm VERY tired.
BTW...If ANY of Ya'll have kids, WATCH OUT for the rotavirus....wash your hands A LOT. Both of my littles have been hospitalized for dehydration (it's gnarly stomach flu.....)
Thanks,
CC

Last edited: Nov 30, 2005
2. Dec 1, 2005

### matt grime

It's just the closure.

If you need to use the definition as given, and you already know that the answer is [0,1] just do it: if x is not in [0,1] show it is not a cluster point, and if x is in [0,1] show it is a cluster point, obviously all points in (0,1) are cluster points, so it only remains to show that 1 and 0 are cluster points which is exactly as hard as knowing that 1/n tends to 0