# Limit Point of a Set

Let $X$ be a topological space with topology $\tau$, and $A$ be a subset of $X$. A point $x \in X$ is said to be the limit point or accumulation point or cluster point of $A$ if each open set containing$x$ contains at least one point of $A$ different from $x$.

In other words, a point $x$ of a topological space $X$ is said to be the limit point of a subset $A$ of $X$ if for every open set $U$ containing $x$ we have

$\left\{ {A \cap U} \right\}\backslash \left\{ x \right\} = \phi$

It is clear from the above definition that the limit point of a set $A$ may or may not be the point of $A$.

Let $X = \left\{ {a,b,c} \right\}$ with topology $\tau = \left\{ {\phi ,\left\{ {a,b} \right\},\left\{ c \right\},X} \right\}$ and$A = \left\{ a \right\}$, then $b$ is the only limit point of $A$, because the open sets containing $b$, namely $\left\{ {a,b} \right\}$ and $X$, also contain a point $a$ of $A$.

On the other hand, $a$ and $b$ are not limit points of $C = \left\{ c \right\}$, because the open set $\left\{ {a,b} \right\}$ containing these points does not contain any points of $C$. Point $c$ is also not a limit point of $C$, because the open set $\left\{ c \right\}$ containing $c$ does not contain any other point of $C$ different from $c$. Thus, the set $C = \left\{ c \right\}$ has no limit points.

As another example, let $X = \left\{ {a,b,c,d,e} \right\}$ with topology $\tau = \left\{ {\phi ,\left\{ a \right\},\left\{ {c,d} \right\},\left\{ {a,c,d} \right\},\left\{ {b,c,d,e} \right\},X} \right\}$. Let $A = \left\{ {a,b,c} \right\}$ then $a$ is not a limit point of $A$, because the open set $\left\{ a \right\}$ containing $a$ does not contain any other point of $A$ different from $a$. $b$ is a limit point of $A$, because the open sets $\left\{ {b,c,d,e} \right\}$ and $X$ containing $b$ also contain a point of $A$ different from $b$. Similarly, $d$ and $e$ are also limit points of $A$. This illustration suggests that a set can have more than one limit point.

Derived Set

Let $\left( {X,\tau } \right)$ be a topological space, and let $A$ be a subset of $X$. The set of all limit points of $A$ is said to be the derived set and is denoted by $D\left( A \right)$ or ${A^d}$. In the above example, $D\left( A \right) = \left\{ {b,d,e} \right\}$.

Remark:

It may be noted that under usual topology, consider the subsets $\left[ {a,b} \right]$, $\left( {a,b} \right)$, $\left[ {a,b} \right)$, $\left( {a,b} \right]$ of real, then all the points of these intervals are limits points.