Give example & prove there is a set with countable infinite set of accumulation point

  • #1

Homework Statement



Give an example and prove there is a set with a countable infinite set of accumulations points.

Homework Equations


An example would be s = {k + 1/n l k element integers, n element natural numbers}

integers are countable infinitie a bijection exists with natural numbers

Def: Let S be a set of real numbers. A, element reals, is an accumulation point iff every neighborhood of A contains infinitely many elements of S.

Def: Let x element reals. Then a set Q, subset Reals, is called a neighborhood of x iff there exists epsilon > 0 such that (x -e, x + e) is a subset of Q.


The Attempt at a Solution



I've spent hours and don't know how to start to prove this. Would appreciate any help!

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
22,089
3,293


Do you know a set that's countable and dense in R? What about that set?
 
  • #3


Rationals are countable and dense in R. Still not sure where I am to take this.
 
  • #4
22,089
3,293


Aren't the rationals a set consisting of all accumulation points?
 
  • #5
Dick
Science Advisor
Homework Helper
26,260
619


Aren't the rationals a set consisting of all accumulation points?
Are you thinking about rationals in Q? If you are thinking about rationals in R, then all points of R are accumulation points. That's not countable. What's wrong with {k+1/n}?
 
  • #6
22,089
3,293


Oh my, it appears I've been reading the questio entirely wrong :blushing:

Yep {k+1/n | k,n naturals} are fine!
 

Related Threads on Give example & prove there is a set with countable infinite set of accumulation point

Replies
1
Views
2K
Replies
3
Views
11K
  • Last Post
Replies
2
Views
7K
  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
11
Views
3K
  • Last Post
2
Replies
29
Views
8K
  • Last Post
Replies
3
Views
2K
Top