# Construct compact set of R with countable limit points

by forget_f1
Tags: compact, construct, countable, limit, points
 P: 11 Construct a compact set of real numbers whose limit points form a countable set.
 Sci Advisor HW Helper PF Gold P: 12,016 Shouldn't a single point be enough?
 Sci Advisor HW Helper PF Gold P: 12,016 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.
 P: 11 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.
 P: 11 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.
 Sci Advisor HW Helper P: 9,397 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.
HW Helper
PF Gold
P: 12,016
 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.
Math
Emeritus