Construct compact set of R with countable limit points

1. Oct 7, 2004

forget_f1

Construct a compact set of real numbers whose limit points form a
countable set.

2. Oct 7, 2004

arildno

Shouldn't a single point be enough?

3. Oct 7, 2004

arildno

Note:
I may have forgotten the precise definition of "the limit point".
You might instead look at a convergent sequence in R; that is a compact set, with one limit point.

4. Oct 7, 2004

forget_f1

for example {(0, 1/n) : n=1,2,3,......} is compact but the only limit point is 0. Still I need countable limit points.

5. Oct 7, 2004

forget_f1

Note: A single point has no limit point, since
a limit point of a set A is a point p such that for any neighborhood of p
(ie Ball(p,r) , where p is the origin and r=radius can take any value >0)
there exists a q≠p where q belongs in B(p,r) and q belongs to A.

6. Oct 8, 2004

matt grime

You can construct a set with one limit point. Now you can make one with two limit points, 3 limit points, indeed any number of limit points countable or otherwise.

7. Oct 8, 2004

arildno

Yeah, I kind of remembered that a bit late...

Finite sets are countable.

Last edited: Oct 8, 2004
8. Oct 8, 2004

HallsofIvy

Staff Emeritus
But the original post probably meant "countably infinite".

9. Oct 8, 2004

forget_f1

Taking A={0, 1/n + 1/m | n,m >=1 in N}. Thus the limit points are 1/n which are countable.
Since the set is closed and bouned then it is compact. (theorem)
Or
It can prove by definition that A is compact, which is what I did since I forgot to use the theorem above which would have made life easier :)