Closedness of $E \cup E'$ 
Definitions: Let $E'$ be the set of all limit points of $E$, and $\bar E=E \cup E'$ the closure of the set $E$. Let $E^c$ be the complement of the set $E$.

Actually, I just want to understand the following assertion:

If $p\notin \bar E$ then $p$ is neither a point of $E$ nor a limit point of $E$, hence $p$ has a neighborhood which does not intersect $E$. $\Longrightarrow$ $\bar E^c$ is open.   

I can see that it imples $N_r(p) \subset E^c$ but to have the above result we also must have $N_r(p) \subset E'^c$ so that $N_r(p) \subset E^c \cap E'^c = \bar E^c$.
 A: $p$ is a limit point of $E$ if and only if every neighborhood containing $p$ contains a point of $E$ that is not equal to $p$. If $p$ is not a point of $E$ and not a limit point of $E$, then negate the definition: there exists some neighborhood containing $p$ that does not contain any points of $E$. This is simply by definition.
A: Note that "if $p \notin \bar{E}$", is equivalent to picking "$p \in \bar{E}^{c}$". The subsequent argument establishes that there is an open neighborhood of $p$ that doesn't intersect $\bar{E}$, or equivalently, it is a subset of $\bar{E}^{c}$ . This means that $\bar{E}^{c}$ is open by definition, as we found an open neighborhood of an arbitrary point in the set that is contained in the set.
A: I assume $E'$ denotes the derived set of $E$, that is, the set of limit points of $E$. I assume also you write $\bar{E}=E\cup E'$ and want to show $\bar{E}$ is closed.
Let $p\notin\bar{E}$, so it belongs neither to $E$ nor to $E'$. In particular, $p$ is not a limit point of $E$, so there is $r>0$ such that the sphere $N_r(p)$ around $p$ with radius $r$ doesn't intersect $E$ at points different from $p$.
On the other hand, $N_r(p)$ cannot intersect $E$ at $p$, because $p\notin E$. Therefore $N_r(p)\cap E=\emptyset$.
Suppose $q\in N_r(p)\cap E'$. Then there exists $s>0$ such that $N_s(q)\subseteq N_r(p)$. As $q\in E'$, $N_s(q)\cap E\ne\emptyset$, which contradicts $N_r(p)\cap E=\emptyset$. Therefore $N_r(p)\cap E'=\emptyset$.
