# Topology: Finding the set of limit points

Determine the set of limit points of:

$$A = { \frac{1}{m} + \frac{1}{n} \in R | m,n \in Z_{+} }$$

I can see that everything less than one can't be reached by this set.
Is my set of limit points (0,1) ?

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If x is in <0, 1>, can x be a limit point of this set? What's the definition of a limit point?

SammyS
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Determine the set of limit points of:

$$A = { \frac{1}{m} + \frac{1}{n} \in R | m,n \in Z_{+} }$$

I can see that everything less than one can't be reached by this set.
Is my set of limit points (0,1) ?
Perhaps you meant to say: Not everything less than one can be reached by this set.

Not sure what you mean by "reach".

From Wolfram Math World:
Limit Point: A number x is a limit point of a set if for all ε>0, there exists a member y of the set different from x such that |y-x|<ε.​

The rational number 3/4 ∈ A, (n=2, m=4), but is 3/4 a limit point of A? Why not?

on the other hand 1/2 is a limit point of A.

Determine the set of limit points of:

$$A = { \frac{1}{m} + \frac{1}{n} \in R | m,n \in Z_{+} }$$

I can see that everything less than one can't be reached by this set.
Is my set of limit points (0,1) ?
Hmm, I think you mean that everything more than one can't be reached by this set. For example, 3/2 is not a limit point of of this set, though it is in the set (take m = 1, n = 2). If that is what you meant, then yes, no points above 1 are limit points of this set.

However, the interval (0,1) is not the answer, either. For example, take any x in (0,1) that is not in A. Then there will always be a neighborhood of x that contains no points of A. Now, let x be a fixed 1/n, in any neighborhood of 1/n, there is always a point of A (different from 1/n); think about why this is. So, right now, I am saying the the points 1/n are all limit points of A. But there is one more than that. Do you know what it is?

It seems as though the limit points may be of the form 1/v, where v is a positive integer..

Yes, points of the form 1/v are limit points, but that is no all of them. There is exactly one more.

and the empty set!

Nope. It is a number.

Wait, what?
We can't approximate the empty set with that set... So it's a limit point, isn't it?

Wait, what?
We can't approximate the empty set with that set... So it's a limit point, isn't it?
A limit point, p, of a set, S, is a point such that every neighborhood of that p contains a point of S different from p. The empty set is not a point. To get the missing limit point, go along with your 1/v idea and think about a REALLY big v.

1/ really big v = zero.. ?

yep, 0 is a limit point of A.

lol, doh. My brain was thinking zero, and my fingers typed empty set.

I'm going to go teach them a lesson, thanks very much!
I'll probably end up posting with other questions from this assignment :P

SammyS
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I'm pretty sure that 0 is NOT a limit point of A because 0 ∉ A.

I'm pretty sure that 0 is NOT a limit point of A because 0 ∉ A.
A limit point does not have to be in the set of which it is a limit point.

For example, consider the set A={1/n:n\in N} then 0 is a limit point because every neighborhood of 0 has a point from A. Thus, 0 is a limit point.

SammyS
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