## construct compact set of R with countable limit points

Construct a compact set of real numbers whose limit points form a
countable set.
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 Recognitions: Gold Member Homework Help Science Advisor Shouldn't a single point be enough?
 Recognitions: Gold Member Homework Help Science Advisor 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.

## construct compact set of R with countable limit points

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

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

Finite sets are countable.

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