What is a limit point

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.

Solution 1:

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.

Solution 2:

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...

Solution 3:

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