# No set (0,1) as accumulation points

• October 7th 2010, 08:04 AM
magus
No set (0,1) as accumulation points
Hello again. My current problem is that I don't understand how this can be:

Show that there is no set with the interval (0,1) as it's accumulation points.

To me it seemes as though the set itself has accumulation points within it so why wouldn't it itself have accumulation points. It's hard to imagine that there is no sequence contained in (0,1) that does not contain an accumulation point in it.

Unfortunately most of the sequences I can think of have an accumulation point of 0 or 1 so I can't think of one specifically.

A useful hint would be appreciated or perhaps a disproof of my assumptions. Thanks.
• October 7th 2010, 08:49 AM
Plato
Quote:

Originally Posted by magus
Show that there is no set with the interval (0,1) as it's accumulation points.

Assume that we are in the space of real numbers with the usual topology.
If $(0,1)$ were the set of accumulation points of a set $A$ would it also be true that $0$ is a limit point of $A$?
• October 7th 2010, 12:08 PM
magus
What exactly is a limit point?

EDIT: Ok so wikipedia tells me it's the same thing as an accumulation point.

Yes it would but I still see no problem with this.
• October 7th 2010, 12:23 PM
Plato
Quote:

Originally Posted by magus
Yes it would but I still see no problem with this.

But the set $(0,1)$ does not contain $0$.
Nevertheless, $0$ is an accumulation point.
So it cannot the set of all accumulation points.
• October 7th 2010, 07:39 PM
magus
Oh ... well that was simple. OK then but wouldn't it be true then for any open interval?
• October 8th 2010, 02:57 AM
Plato
Quote:

Originally Posted by magus
Oh ... well that was simple. OK then but wouldn't it be true then for any open interval?

Yes that is true. I think that was the point of the problem.
In the real numbers, the derived set (set of all accumulation points) of a set is closed.