Accumulation Points

Definition: Accumulation Point

Let $(X,\tau )$ be a topological space and let $S$ be a subset of $X$.

A point $p\in X$ is an accumulation point of $S$ iff every neighbourhood of $p$ contains a point of $S$ different from $p$.

NOTE

Limit points are also known as limit points or cluster points.

EXAMPLE

Consider the topological space $(X,\tau )$, where $X=\{a,b,c,d,e\}$ and $\tau =\{\varnothing ,X,\{a\},\{c,d\},\{a,c,d\},\{b,c,d,e\}\}$. Suppose $S=\{a,b,c\}$.

The set $\{a\}$ is open but it does not contain any other element of $S$, hence $a$ is not a limit point of $S$.

The sets which contain $b$ are $X$ and $\{b,c,d,e\}$. We see that $X$ contains $a$ (or alternatively $c$) which is another point of $S$. Similarly, we see that $\{b,c,d,e\}$ contain $c$ which is again another point of $S$. Therefore, all sets which contain $b$ also contain another point of $S$ and so $b$ is a limit point of $S$.

The set $\{c,d\}$ contains $c$ but the only other element it contains is $d$ which is not an element of $S$. Therefore, $c$ is not a limit point of $S$.

The sets which contain $d$ are $X$, $\{c,d\}$, $\{a,c,d\}$ and $\{b,c,d,e\}$. The set $\{c,d\}$ also contains $c$ which is a point of $S$. The set $\{a,c,d\}$ contains both $a$ and $c$ which are points of $S$. The set $\{b,c,d,e\}$ contains $b$ and $c$ which are points of $S$. Hence, $d$ is a limit point of $S$ even though $d$ is not in $S$.

The sets which contain $e$ are $X$ and $\{b,c,d,e\}$. Both contain multiple elements which belong to $S$ and thus $e$ is a limit point.