Theorem 2.27: If $X$ is a metric space and $E \subset X$, then $\bar E$ (the closure of $E$)
is closed.
The proof says: If $p \in X$ and $p \not \in \bar E$ then $p$ is neither a point of $E$ nor a limit point of $E$. Hence $p$ has a neighborhood which does not intersect $E$. The complement of $\bar E$ is therefore open. Hence $\bar E$ is closed.
I'm particularly questioning about ''Hence $p$ has a neighborhood which does not intersect $E$. The complement of $\bar E$ is therefore open.'' Should we also prove that the neighborhood of $p$ also does not intersect $E'$ (the set of all limit points of $E$)?
Here's what I tried to prove, by contrapositive: ''For any $p \in {\bar E}^c$, if $N_r(p) \cap E' \ne \emptyset$ then $N_r(p) \cap E \ne \emptyset$''.
Proof: For any $p \in {\bar E}^c$, if $N_r(p) \cap E' \ne \emptyset$, then take $q \in N_r(p) \cap E'$, $\exists N_h(q)$ s.t. $N_h(q) \subset N_r(p)$. Since $q \in E'$ is a limit point, $N_h(q) \cap E \ne \emptyset$, and hence $N_r(p) \cap E \ne \emptyset$.
I'm not quite sure whether this is necessary. Or is there anything I missed from Rudin's proof?
Best Answer
$p$ was chosen randomly, so could be any point not in $E$, not in $E'$, as any such $p$ was explicitly ruled out as a limit point of $E$, hence the existence of a neighborhood of $p$ will not intersect $E'$.
Your proof is fine. You explained nicely why Rudin's claim follows.
Rudin is notorious for leaving some of the "links" between "stepping stones" (steps in his proofs) "to the reader". You've just filled in some of those unwritten details, and you are correct in those details. It's always a good idea to do so as you read any text, when anything is not immediately apparent to you while reading through a proof. In particular, that can often be the case when reading Rudin. When you revisit the proofs, then, that "extra work" will pay off, as you'll have made the connections, and will then be able to reread Rudin's proofs and follow them without so much effort.
Sometimes filling in the details may amount to simply "unpacking" the definition(s) of the terms being used in a theorem (what it means to be a limit point, e.g.). You'll find that's a sound way to learn the definitions inside-out, and to review theorems when they are used in subsequent proofs. (In all honesty, I personally "wrote, expanded upon, rewrote, extended, rewrote again" virtually all of Baby Rudin when I first encountered the text.