Understanding the idea of a Limit Point (Topology)

I have attached an image of how I was visualizing a limit point, but I'm now not so sure that I have understood the concept correctly after attempting to really draw out what I was visualizing.

I'll mention the definition of Neighbourhood and Limit Point from Rudin's Analysis just for a refresher:

Definitions

Let $X$ be a metric space. All points and sets mentioned below are understood to be elements and subsets of $X$. A neighbourhood of $p$ is a set $N_{r}(p)$ consisting of all $q$ such that $d(p,q)<r$, for some $r>0$. A point $p$ is a limit point of the set $E$ if every neighbourhood of $p$ contains a point $q \neq p$ such that $q \in E$

I ran into a roadblock understanding the boundary, closure, and interior of halfspaces bounded by hyperplanes, and I think it runs back down to my misunderstanding of the limit point. Here is the figure I created:

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EDIT: New figure for a limit point $p$ Please let me know if this is more accurate!

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I am imagining this as happening at the "infinitesimal" level, so for example, $r_{1}$ is not actually as large as shown in the figure. The three general operations I imagine happen are:

$$d(r_{n},r_{n-1}) \rightarrow 0$$ $$\forall r \in \mathbb{R}, \exists q \in E \text{ }|\text{ } d(p,q_{n})<r_{n}, n \in \mathbb{N}$$ $$d(p,q_{1}) \rightarrow 0$$

So I think of every neighbourhood around the point $p$ as being circles of expanding radius, and have the condition that there must be some $q \neq p$ where $q \in N_{r}(p)$ and I think of this as being a condition for each radius. Is this an overcomplication? I realized that for say some $r_{1}$ that if there exists a $q \in E$ so that $d(p,q) < r_{1}$, then for any $r_{i} > r_{1}$, that same point $q$ which worked for the neighbourhood $N_{r_{1}}(p)$ will work for the neighbourhood $N_{r_{i}}(p)$. So, are these three operations sort of additional constraints on the situation that do not add any relevant information?

Main Question

My problem now, assuming that the condition is that for all $r \in \mathbb{R}$, there is some $q \in E$ so that $d(p,q) < r$, for an arbitrarily small $r>0$, then how can I visualize this? How can two points $p$ and $q$ where $p \neq q$ actually be distinct when I can shrink my radius arbitrarily small? If anyone could give me a concrete example or an explanation of where I went wrong/what could clear things up for me that would be very helpful.

Thank you!

Solution 1:

The problem seems to be that you’ve got your quantifiers backwards. It isn’t that there’s a single $q\in E$ such that $d(p,q)<r$ for arbitrarily small $r$. Rather, for each positive $r$, no matter how small, there is a $q\in E$ such that $d(p,q)<r$. The choice of $q$ depends on $r$. In symbols, this is the difference between

$$\exists q\in E\setminus\{p\}\;\forall r>0 (d(p,q)<r$$ and $$\forall r>0\;\exists q\in E\setminus\{p\}(d(p,q)<r\;.$$

For a simple example, let $$E=\left\{\frac1n:n\in\mathbb{Z}^+\right\}\;,$$ and let $p=0$. For $r=0.1$, you can take for $q$ any $1/n$ with $n>10$. With $r=0.01$, on the other hand, you’ll need to choose a $1/n$ with $n>100$. And so on.

I’d also say that your picture is inside-out: you should think of circles of decreasing radius squeezing in closer and closer to $p$. Then $p$ is a limit point of $E$ if within each of those circles, no matter how close to $p$, there is at least one point of $E$ different from $p$ itself.

Added:

Now let’s take a look at your three ‘general operations’.

$$d(r_{n},r_{n-1}) \to 0$$

If you choose a sequence $\langle r_n:n\in\mathbb{N}\rangle$ converging to $0$, then it will automatically be the case that $d(r_{n},r_{n-1}) \to 0$ as $n\to\infty$, but this is a side-effect, not something on which you should focus. What’s important is that $r_n\to 0$ as $n\to\infty$; if that’s the case, and if for each $n\in\mathbb{N}$ you have a $q_n\in E$ such that $q_n\ne p$ and $d(p,q_n)<r_n$, then $p$ is a limit point of $E$.

$$\forall r \in \mathbb{R} \exists q \in E \big(d(p,q_{n})<r_{n}, n \in \mathbb{N}\big)$$

As written, this doesn’t make sense: how are the single $r$ and $q$ in the quantifiers related to the $r_n$ and $q_n$ in the quantified statement? You could correctly write any of the following, since all of them say that $p$ is a limit point of $E$:

$$\begin{align*} &\forall r>0 \exists q(r)\in E\big(q(r)\ne p\text{ and }d(p,q(r))<r\big)\\ &\forall n\in\mathbb{Z}^+\exists q_n\in E\left(q(r)\ne p\text{ and }d(p,q_n)<\frac1n\right)\\ &\forall n\in\mathbb{N}\exists q_n\in E\left(q(r)\ne p\text{ and }d(p,q_n) < \frac1{2^n}\right) \end{align*}$$

Indeed, if $\langle r_n:n\in\mathbb{N}\rangle$ is any sequence of positive real numbers converging to $0$, you could take

$$\forall n\in\mathbb{N}\exists q_n\in E\big(q(r)\ne p\text{ and }d(p,q_n) < r_n\big)$$

as your definition of ‘$p$ is a limit point of $E$’.

$$d(p,q_{1}) \to 0$$

As written this makes no sense, since $p$ and $q_1$ are single, fixed points: there is no sequence here. Did you mean $d(p,q_n)\to 0$? That isn’t enough as it stands, because it says nothing about the nature of the $q_n$. What does work is this:

A point $p$ is a limit point of a set $E$ if and only if there is a sequence $\langle q_n:n\in\mathbb{N}\rangle$ of points of $E\setminus\{p\}$ such that $d(p,q_n)\to 0$ as $n\to\infty$.

(This of course assumes that there is a metric $d$; this definition doesn’t work for topological spaces in general.)