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Wikipedia seems to describe the topic with extreme complexity for me.

In mathematics, a limit point of a set $S$ in a topological space $X$ is a point $x$ (which is in $X$, but not necessarily in $S$) that can be "approximated" by points of $S$ in the sense that every neighbourhood of $x$ with respect to the topology on $X$ also contains a point of $S$ other than $x$ itself. Note that $x$ does not have to be an element of $S$.

I don't understand the relationship between $S$ and $X$ and what it means for $x$ to be a limit point.

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    $\begingroup$ $p$ is a limit point for a set $S$ if for each $U$ open set in $X$, for which $p\in U$, the set $(U\setminus\{p\})\cap S\neq\varnothing$. $\endgroup$
    – janmarqz
    Feb 4, 2014 at 20:56
  • $\begingroup$ S is the set we are talking about, X is the space. S is in X. $\endgroup$
    – ahron
    Dec 4, 2021 at 15:38

3 Answers 3

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The definition that you shared from the Wikipedia article defines a limit point in the most abstract (generalized) way, which is important; but this isn't the best definition for building an initial understanding.

In most cases, there exists a distance that allows us to say how far points are from each other (we say that the space is metric). For instance on $\mathbb R$, the distance between $x$ and $y$ is $|x-y|$.

Now a limit point of a set $S$ is a point which has points of $S$ other than itself, arbitrarily close to it. A non-trivial example is that $0$ is a limit point of $[0,1]$, because it can be approximated by points of the form $\frac1n$ for $n\in\mathbb N^*$.

More formally, $x$ is a limit point of $S$ if for all $\epsilon>0$, there is a point $y\in S\setminus\{x\}$ with $d(x,y)<\epsilon$.

Wikipedia uses the notion of topological space to be more general than metric spaces, but this notion is needed only for advanced math. For starters, it suffices to know that any metric space is in particular a topological space, and that in this case we can use the distance to define limits.

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  • $\begingroup$ Appreciate your help but you have added an ϵ which you haven't previously defined. Can you please clarify? $\endgroup$
    – iordanis
    Feb 4, 2014 at 20:45
  • $\begingroup$ the epsilon is introduced with a "for all" quantifier. It formalizes the fact that poinst of $S$ are "arbitrarily close" to $x$, because for each distance $\epsilon$, no matter what it is, there is a point of $S$ which is close to $x$ with respect to $\epsilon$ (their distance is $<\epsilon$) $\endgroup$
    – Denis
    Feb 4, 2014 at 20:47
  • $\begingroup$ Also you say a limit point is any point that has an arbitrary distance from all other S points. That sounds extremely complicated to me and ill-defined. Can you please clarify? $\endgroup$
    – iordanis
    Feb 4, 2014 at 20:47
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    $\begingroup$ @Denis you say "all points of S are limits point of S". But I don't think this is true in general - isolated points of S are not limit points. $\endgroup$
    – TSJ
    Feb 15, 2015 at 23:17
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    $\begingroup$ @Denis: Perhaps if you use the clever "let me google that for you" link provided, you will see that the top result is the Wikipedia article on "limit point" where a limit point is defined so that isolated points are not limit points. In particular, while the closure of $S$ contains all points of $S$, a limit point $x$ is required to have all neighborhoods intersect $S$ in a point other than $x$. $\endgroup$
    – hardmath
    May 7, 2015 at 12:33
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A limit point of a set $S$ in a metric space $X$ can be either in $S$ or in $X$.

So you have this point, call it $x$. Now think of an arbitrary distance (using your distance metric, $d(x,y)$). Call it $r$. If $x$ is a limit point, then it means that for ANY $r>0$, you will be able to find some other point $y \in S$ such that $d(x,y) <r$. You can make $r$ arbitrarily small, but within the distance $r$, our point $x$ will not be alone. Its neighborhood will never be empty: a neighborhood is just the set of points that are within the distance $r$ from $x$.

This essentially means that for any neighborhood of $x$ that we select, there is an infinite number of points in the neighborhood, 'accumulated around' x. Example:

Consider the set $\mathbb{R}^1$, the real line. This is our metric space with the usual Euclidean Norm for distance. Now consider: $$ S = \{1 + \frac1n \mid n \in \mathbb{N}\}$$ This set $S$ has a limit point at $1 \in \mathbb{R}^1$ (and only at 1) because for any $r>0$, we can always find some $n \in \mathbb{N}$ such that $1+ \frac1n < 1+r$, so that there is some $s \in S$ with a distance to $1$ that is less than $r$.

Intuition: you can then expand this notion: in $\mathbb{R}^2$, we think of $r$ as a radius, and the neighborhood is a circle if you draw the set. In $\mathbb{R}^3$, $r$ becomes a sphere (ball), etc...

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    $\begingroup$ can the limit points be considered graphically at the edge of a set? $\endgroup$
    – iordanis
    Feb 4, 2014 at 22:00
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    $\begingroup$ Not quite. For example, let $\mathbb{R}^1$ be your metric space. Then $\mathbb{Q} \subset \mathbb{R}^1$. What are the limit points of $\mathbb{Q}$? Well, every $x \in \mathbb{R}$ is a limit point of $\mathbb{Q}$. (I won't give the proof here, look it up if you're interested.) If you consider the sets graphically, then they're not at the 'edge' of a set: they're inside the set. However, in many cases, the limit point is at the 'edge' of the set (so if you are trying to find a limit point, you would not be wrong to look there first). $\endgroup$
    – Newb
    Feb 4, 2014 at 22:11
  • $\begingroup$ @Newb could you give an example where the limit point is at the edge? I think in many cases, such as a interval in $\mathbb{R}^1$ or common shapes in $\mathbb{R}^2$ (such as a filled circle), the limit points consist of every interior point as well as the points on the "edge". $\endgroup$
    – TSJ
    Feb 15, 2015 at 23:20
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    $\begingroup$ @TSJ That's right -- it's usually just the "edge points" that people have trouble understanding. I think the example I gave for $\mathbb{R}^1$ is an "edge" example. (This graphical analogy breaks down when inspected thoroughly, of course -- it's just meant to provide intuition.) $\endgroup$
    – Newb
    Feb 15, 2015 at 23:32
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A point $a$ is said to be a limit point of a set $S$ if there are points in $S$ other than $a$ that are arbitrarily close to $a$ but never become equal to $a$. For example $1$ is a limit point of the intervals $[0,1]$ and $[0,2]$ because $\{0.9,0.99,0.999,0.999 \dots\}$ is a sequence of points in those intervals that approaches $1$ but never become equal to $1$.

Source: Limits, Continuity and Differentiability - David Levermore

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  • $\begingroup$ Would this imply that every point in the set $[0,2]$ is a limit point, since you can get arbitrarily close to any of them? $\endgroup$
    – Alex
    Feb 18, 2021 at 12:46
  • $\begingroup$ arbitrarily close to a but never become equal to a. For points in [0,2] in real numbers, one can both get arbitrarily close to them and also become equal to them. So no, every point is not a limit point. Now, suppose we consider the set S [0,2] in the space of rational numbers Q. Now the points in S can get arbitrarily close to the point denoting square root of 2 but never become equal to it. $\endgroup$
    – ahron
    Dec 4, 2021 at 15:56

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