Real analysis question: accumulation point

[R.M.]

In my real analysis text, I have read, reread, re-re-read, and re-re-re-read the definition of an accumulation point. Then after letting my thoughts marinate for awhile, I read it again.

Definition from text: "A point p is an accumulation point of a set S if it is the limit of a sequence of points of S - {p}. It is equivalent to require that every ball (p - r, p + r) about p intersect S - {p}."

One of my fellow classmates gave me the following advice: 'stand on a point and stretch out your arms. What do you hit?' I think that was the missing puzzle piece I needed to put together the following:
1. A "ball" is just a range of points. In 1-D space, a ball would be an interval; in 2-D space, a ball would be a circle, etc... So then r would be the 'radius' of my interval or circle or sphere or ....... depending on the space I'm in.

2. If I have a set of points, and I'm standing on a point somewhere, I should be able to create any size ball and include at least one point in the original set, NOT INCLUDING the point I'm standing on.

3. Example: I'm standing on a number line, my set S is the rational numbers, and I stand on the point 4. I can create an interval (ball) that hits at least one rational number, therefore 4 is an accumulation point. I can also stand on the point 4.1564654 and do the same thing, therefore 4.1564654 is an accumulation point, as is every single real number. In fact, I can ALWAYS hit a rational with any size interval/ball by standing on ANY real number. Therefore all the Real Numbers are accumulation points.

4. Another example: I'm still standing on a number line, my set S is the integers, and I stand on the point 4. I create an interval that is 0.13 on either side of 4, hitting no other integer. I cannot include 4, because I'm standing on it. Therefore 4 is not an accumulation point. In fact, there are no accumulation points, because I MAY hit an integer with my interval, but I cannot GUARANTEE I will hit an integer.

Am I understanding accumulation points correctly? Thanks in advance for any assistance.

 

[Steven G]

Looks good to me.

Just understand that the ball needs to be small to confirm that a point is a limit point (or accumulation point).

Which textbook are you using?

You have the right approach about looking at the definition a number of times and thinking about what it could possible mean.

 

[pka]

Definition from text: "A point p is an accumulation point of a set S if it is the limit of a sequence of points of S - {p}. It is equivalent to require that every ball (p - r, p + r) about p intersect S - {p}."

Although there are slight differences in wording the all metric spaces a point \(p\) is an accumulation point of a set \(S\) provided any open set that contains \(p\) also contains a point of \(S\) distinct from \(p\). That means that no matter how small a neighborhood, \(\mathcal{N}\) of \(p\) we choose that is always a point \(q\ne p,~\&~q\in\mathcal{N}\). In other words, there is always a point of \(S\) not \(p\) close to \(p\).

 

[Dr.Peterson]

The one place I notice where you need to be a little more careful is here:

3. Example: I'm standing on a number line, my set S is the rational numbers, and I stand on the point 4. I can create an interval (ball) that hits at least one rational number, therefore 4 is an accumulation point. I can also stand on the point 4.1564654 and do the same thing, therefore 4.1564654 is an accumulation point, as is every single real number. In fact, I can ALWAYS hit a rational with any size interval/ball by standing on ANY real number. Therefore all the Real Numbers are accumulation points.

It's not that there is some such ball ("I can create one"), but that any ball hits a number ("I can't avoid hitting something").

To extend the nice illustration of stretching your arms out, no matter how much you diet, you're never thin enough to avoid having a point in the set inside you! (And therefore there will be infinitely many such points inside you.)

I could also quibble by pointing out that 4.1564654 is a rational number, so it's not a really good example. I imagine you had in mind some irrational number 4.1564654... .

But I think you have the ideas. Standing inside a definition and looking (or feeling) around is a good way to get familiar with it.

 

[Steven G]

Here is a great problem (at least in my opinion) for you to consider.

Suppose there is a set S such that any ball around point p contains a point of S, different from p (p may not even be in S)

Is the set S finite or infinite? Of course state the proof.

 

[pka]

Here is a great problem (at least in my opinion) for you to consider.
Suppose there is a set S such that any ball around point p contains a point of S, different from p (p may not even be in S)
Is the set S finite or infinite? Of course state the proof.

I hope that you are working in as infinite metric space and not a finite topological space.
Lets say the space is \(\mathbb{R}^2\).

 

[Steven G]

I hope that you are working in as infinite metric space and not a finite topological space.
Lets say the space is \(\mathbb{R}^2\).

You are correct. I should have stated the space. Thanks!

 

[R.M.]

Thanks everyone for the clarifications! The textbook for the class I'm taking is "Real Analysis" by Frank Morgan.