Are Two Definitions of Limit Points the Same?

In summary, there are two definitions of limit points, but they are in fact equivalent. A point x in a metric space is a limit point if every neighborhood of x contains at least one element in the metric space not equal to x. This is the same as saying x is in the closure of A if it lies in the closure of A - {x}. While there may be different ways to define the closure, the two commonly used definitions are A union A', or the intersection of all closed sets containing A.
  • #1
ehrenfest
2,020
1

Homework Statement


I have seen two definitions of limit points. Are they the same:

1)x is a limit point of a set A in X iff each nbhd of x contains a point of A other than x

2) x is a limit point of A if it lies in the closure of A - {x}


Homework Equations





The Attempt at a Solution

 
Physics news on Phys.org
  • #2
No, not quite. The second is not a definition of a limit point.

A point x in a metric space is said to be a limit point if every neighborhood of x contains at least one element in the metric space not equal to x.

Consider A', the set of all limit points of A. The closure of A is A U A'.
 
  • #3
Actually, I just realized that those two definitions are the same for the following reason:

x is in the closure of A-{x}
iff
x is in every closed set containing A-{x}
iff
there does not exist a nbhd U_x of s.t. [tex] U_x \cap (A-{x}) = \emptyset [/tex]
 
  • #4
Am I right?
 
  • #5
If you're right, you'll be able to prove it.

You're just making everything exponentially more complicated than it actually is. True mathematicians aim for simplicity.
 
  • #6
varygoode said:
If you're right, you'll be able to prove it.

You're just making everything exponentially more complicated than it actually is. True mathematicians aim for simplicity.

What are you talking about? I am asking if my proof in the third post makes sense.
 
  • #7
Someone, please, am I right?
 
  • #8
It looks OK, but you're somewhat over-complicating it. Try to use the fact that y is in the closure of B iff every nbhd of y intersects B.
 
  • #9
morphism said:
It looks OK, but you're somewhat over-complicating it. Try to use the fact that y is in the closure of B iff every nbhd of y intersects B.

Then the equivalence of those two definitions is immediate, isn't it? I really do not understand why two people have said I am overcomplicating this...
 
  • #10
ehrenfest said:
What are you talking about? I am asking if my proof in the third post makes sense.

I was saying if it makes sense, it has a proof. So if you can't come up with a solid proof, there's a higher chance it doesn't make sense.

ehrenfest said:
Then the equivalence of those two definitions is immediate, isn't it? I really do not understand why two people have said I am overcomplicating this...

But see, in your second "definition" you talk about the closure of a set A. But the closure is defined as the set A unioned with the set of all of A's limit points. And then you still need a definition for a limit point. So it comes down to the fact that you want to use the term in the definition, which just further complicates things. That's what I mean.
 
  • #11
The closure of A is defined as the intersection of all closed sets containing A. It is equivalently A union A'. Both of these definitions are commonly found in the literature and indeed, they are the seem. I think that you are complicating things and that you should make sure you know more about the topic before you say that something is or is not a definition.
 

1. What is a limit point?

A limit point is a point in a set that can be approached arbitrarily closely by other points in the set. In other words, any neighborhood of a limit point contains infinitely many other points from the set.

2. What are the two definitions of limit points?

The first definition of a limit point is a point x in a set S such that every neighborhood of x contains at least one point from S other than x. The second definition is a point x in a set S such that for every epsilon greater than 0, the neighborhood (x-epsilon, x+epsilon) contains infinitely many points from S.

3. Are these two definitions equivalent?

Yes, the two definitions of limit points are equivalent. This means that a point x in a set S satisfies one definition if and only if it satisfies the other definition.

4. Can you provide an example of a limit point?

Yes, consider the set S = {1/n | n is a positive integer}. The point 0 is a limit point of this set because every neighborhood of 0 contains infinitely many points from S (specifically, 1/n for all sufficiently large n).

5. Why are limit points important in mathematics?

Limit points are important in mathematics because they help define concepts such as continuity and compactness. They also play a crucial role in the foundations of calculus and real analysis.

Similar threads

  • Calculus and Beyond Homework Help
Replies
5
Views
849
  • Calculus and Beyond Homework Help
Replies
12
Views
695
  • Calculus and Beyond Homework Help
Replies
10
Views
792
  • Calculus and Beyond Homework Help
2
Replies
58
Views
3K
  • Calculus and Beyond Homework Help
Replies
9
Views
712
  • Calculus and Beyond Homework Help
Replies
3
Views
914
  • Calculus and Beyond Homework Help
Replies
16
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
707
  • Calculus and Beyond Homework Help
Replies
4
Views
858
  • Calculus and Beyond Homework Help
Replies
4
Views
775
Back
Top