Are Two Definitions of Limit Points the Same?

[ehrenfest]

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

 

[varygoode]

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'.

 

[ehrenfest]

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. $$U_x \cap (A-{x}) = \emptyset$$

 

[ehrenfest]

Am I right?

 

[varygoode]

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.

 

[ehrenfest]

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.

 

[ehrenfest]

Someone, please, am I right?

 

[morphism]

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.

 

[ehrenfest]

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...

 

[varygoode]

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.

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.

 

[ehrenfest]

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.