Construct compact set of R with countable limit points

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Discussion Overview

The discussion revolves around the construction of a compact set of real numbers that has countable limit points. Participants explore definitions and examples related to limit points, compactness, and the nature of countable sets.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant suggests constructing a compact set with countable limit points.
  • Another participant questions whether a single point could suffice as a limit point.
  • A participant notes that a convergent sequence in R is a compact set with one limit point, potentially misunderstanding the requirement for countable limit points.
  • One example provided is the set {(0, 1/n) : n=1,2,3,...}, which is compact with a single limit point at 0, but does not meet the requirement for countable limit points.
  • There is a clarification about the definition of limit points, emphasizing that a single point does not qualify as a limit point.
  • Another participant asserts that it is possible to construct sets with multiple limit points, including countably infinite limit points.
  • A participant proposes the set A={0, 1/n + 1/m | n,m >=1 in N}, claiming that the limit points are countable and that the set is compact.

Areas of Agreement / Disagreement

Participants express differing views on the nature of limit points and the requirements for constructing a compact set with countable limit points. There is no consensus on a specific construction that meets the criteria.

Contextual Notes

Some participants exhibit uncertainty regarding the definitions of limit points and compactness, and there are references to theorems that are not fully explored or applied in the discussion.

forget_f1
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Construct a compact set of real numbers whose limit points form a
countable set.
 
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Shouldn't a single point be enough?
 
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.
 
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.
 
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.
 
forget_f1 said:
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...:redface:

Finite sets are countable.
 
Last edited:
arildno said:
Yeah, I kind of remembered that a bit late...:redface:

Finite sets are countable.

But the original post probably meant "countably infinite".
 
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 :)
 

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