Register to reply

Construct compact set of R with countable limit points

by forget_f1
Tags: compact, construct, countable, limit, points
Share this thread:
forget_f1
#1
Oct7-04, 07:43 PM
P: 11
Construct a compact set of real numbers whose limit points form a
countable set.
Phys.Org News Partner Science news on Phys.org
Hoverbike drone project for air transport takes off
Earlier Stone Age artifacts found in Northern Cape of South Africa
Study reveals new characteristics of complex oxide surfaces
arildno
#2
Oct7-04, 07:48 PM
Sci Advisor
HW Helper
PF Gold
P: 12,016
Shouldn't a single point be enough?
arildno
#3
Oct7-04, 08:04 PM
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.

forget_f1
#4
Oct7-04, 10:07 PM
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.
forget_f1
#5
Oct7-04, 10:11 PM
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.
matt grime
#6
Oct8-04, 03:55 AM
Sci Advisor
HW Helper
P: 9,398
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.
arildno
#7
Oct8-04, 04:58 AM
Sci Advisor
HW Helper
PF Gold
P: 12,016
Quote 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.
HallsofIvy
#8
Oct8-04, 06:51 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,323
Quote Quote by arildno
Yeah, I kind of remembered that a bit late...

Finite sets are countable.
But the original post probably meant "countably infinite".
forget_f1
#9
Oct8-04, 09:35 AM
P: 11
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 :)


Register to reply

Related Discussions
Compact-valued range doesnot imply compact graph Calculus 3
Limit Points... Calculus 4
Limit points Calculus & Beyond Homework 10
A metric space having a countable dense subset has a countable base. Calculus & Beyond Homework 2
Countable But Not Second Countable Topological Space General Math 3