Let is consider a topological space $(X, \tau) $ and $A\subset X$ .
$\textbf{Definition}$: $x\in X$ is said to be an accumulation point of $A$ if $\forall U\in\tau$ with $x\in U$ satisfy $A\cap U\setminus \{x\}\neq \emptyset$
In other words every open sets containing $x$ contains $\color{red}{a \space point}$ of $A$ other than $x$.
Consider two examples:
Consider $(0,1)\subset(\Bbb{R}, \tau_{std}) $.Then any open interval $0\in (-\delta, \delta) $ contains infinitely many points of $(-\delta, \delta) $
Let $X=\{a,b\}$ and $\tau=\{\emptyset,\{a\},\{a,b\},\{a,b,c\}\}$
Let $A=\{a,b\}$
Then $c$ is a limit point of $A$ as any open set containing $c$ $[$infact there is only one such open set , $\{a, b, c\}]$ contains a point other than $c$ i.e $\{a, b, c\}\cap A\setminus \{c\}=\{a, b\}$
Note: $b$ is also limit point of $A$ but $a$ is not a limit point of $A$ (verify)
In example $1$ , any open set containing the limit point contains infinitely many elements whereas in example $2$ , we have seen that open set containing the limit point is finite. Why?
$\textbf{Definition}$ : A topological space $(X, \tau) $ is said to a $T_1$-space if $\forall x\neq y\in X$ , there exists open sets $U\ni x$ and $y\ni V$ such that $y\notin U$ and $x\notin V$.
The following conditions are equivalent :
$(X, \tau) $ is $T_1$ -space.
$\forall x\in X ,\{x\}$ is closed.
$A\subset X$ is closed whenever $A$ is finite.
$\textbf{Examples}$ : $(\Bbb{R}, \tau_{std}) $, any metric space, $(X, \tau_{\text{cofinte}}) $, $(X, \tau_{\text{cocountable}}) $ etc.
$\textbf{Non Examples}$: $(X, \tau_{\text{indiscrete}}) $, $(X=\{a, b\}, \tau=\{\emptyset,\{a\},\{a,b\}\}) $ ,
$(X=\{a,b\},\tau=\{\emptyset,\{a\},\{a,b\},\{a,b,c\}\})$ etc.
$\textbf{Theorem}$: Let $(X, \tau) $ be a $T_1$-space and $A\subset X$ . Then $x\in X$ is a limit point of $A$ . Then any open set containing $x$ contains an infinite subset of $A$ .
Proof: Suppose there is an open set $U$ containing $x$ such that $A\cap U\setminus \{x\}$ is finite. Then $X\setminus (U\setminus \{x\})$ is an open set containing $x$ which doesn't contain any point of $A$ . Hence it contradict that $x$ is a limit point of $A$.
Some interesting questions related to topology.