From definition, if $X$ is a metric space, if $E \subset X$, and if $E'$ denotes the set of all limit points of $E$ in $X$, then the closure of $E$ is the set $\overline{E}=E \cup E'$.
I need to prove that $\overline{E}$ is closed. By a closed set, I mean that all limit points of a set are in the same set. A limit point of a set is a point whereby every neighbourhood of the point contains a $q$ such that $q$ is an element of the set.
The proof provided in Theorem 2.27(a) of Rudin's Principles of Mathematical Analysis is as follows:
If $p \in X$ and $p \notin \overline{E}$ then $p$ is neither a point of $E$ nor a limit point of $E$. Hence $p$ has a neighbourhood which does not intersect $E$. The complement of $\overline{E}$ is therefore open. Hence $\overline{E}$ is closed.
My question is with how "Hence $p$ has a neighbourhood which does not intersect $E$." leads to "The complement of $\overline{E}$ is therefore open.".
To show that the complement of $\overline{E}$ is open, we need to show that $p$ has a neighbouthood which does not intersect $\overline{E}$, not just $E$. Unless it is true that if the neighbourhood (an open set, since neighbourhoods are all open sets) does not intersect $E$, then it also does not intersect $E'$. Is that true?
Best Answer
Your definition of limit point is a little off: it should say that $p$ is a limit point of a set $E$ if every nbhd of $p$ contains a point $q\in E$ such that $q\ne p$. Without that last condition $2$ would be a limit point of $[0,1]\cup\{2\}$ in $\Bbb R$, which is not what we want the term to mean.
You’ve started with a point $p\in X\setminus\overline E$, meaning that $p\notin E$ and $p\notin E'$. Since $p\notin E'$, $p$ has a nbhd $U$ that contains no point of $E$ different from $p$. Since $p\notin E$, this means that $U$ contains no point of $E$ at all: $U\cap E=\varnothing$. Suppose that $x\in U\cap E'$; then $U$ is a nbhd of $x$, and $x\in E'$, so $U\cap E\ne\varnothing$, which we just saw is not the case. Thus, $U\cap E'=\varnothing$, and therefore $U\cap\overline E=\varnothing$, as desired.