E is closed if every limit point of E is a point of E?
Should that be "E is closed if every point of E, is a limit point"?
I don't understand. Limit points are essentially points that hug other points. How can a limit point of E, not be a point of E?
On
The definition is correct as is: "$E$ is closed if every limit point of $E$ is a point of $E$."
To see how a limit point could not be a point of $E$, consider the (open) interval $E=(0,1)$. Then for any $\epsilon$ neighborhood of zero, there is some positive number ($\min\left(\frac{\epsilon}{2},\frac{1}{2}\right)$ for instance) which is contained in the neighborhood and $E$, but $0$ itself is not in $E$. Hence $E$ is not closed.
On
$E$ is closed but no need every point is a limit point, say $E=\{0\}$, it has no limit point but it's closed in $\mathbb{R}$
The closure of $E$, in a topological space $X$ containing it, is $E$ together its limit point. This proves your statement: "E is closed if every limit point of E is a point of E" which is true.
Conversely, a closed subset of a topological space can have no limit points: for example $\{0\}$ is closed in $\mathbb R$, but it hasn't limit points; it has only isolated point and it is called discrete.
Closed subspaces such that "every point of E is a limit point" are called perfect.
Finally, $E=\{\frac 1n:n\in\mathbb N-\{0\}\}$ has $0$ as limit point and $0\notin E$; note that $E$ is not closed in $\mathbb R$.