If $E$ is open set, then every limit point of $E$ need not to be in $E$

Question

the set $E$ is said to be open set if every element $x$ in $E$ then $x$ is interior point in $E$, for example the open set (interval) $$I=(a,b)$$ the limit point $x$ of a set $E$ if there is a real number $r\gt 0$ $$ ( N_r(x)-{x})-\ E^c\neq \emptyset $$ the set of all limit points of $I$ is the closed set (interval) $$I'=[a,b]$$ then $a$ is limit point of $I$ but $a$ is not in $I$ , thus don't every limit point of a set $E$ belong to $E$ , if was this case satisfied then every open set is closed set and the visa versa. I need more information about this discussion. and thanks advance