In 'Topology' by Munkres, he leaves as an exercise to prove that $\mathbb{R}^J$ is not normal under the product topology when $J$ is uncountable. The proof outlined as exercise 32.9 is the same one given in the 'Counterexamples in Topology' book, space #103, Uncountable Products of $\mathbb{Z}^+$. My question is a two-fer:
1) Is what I've outlined below an alternative proof to parts (c) and (d) in his problem as stated? Or have I gone wrong at some step? I am skeptical, as my "proof" is quite a bit simpler than what is given in the outline in the book, and it is mentioned a few times that the proof is quite difficult. Perhaps I am missing the point, which is to see what Stone did in his proof with his particular sequences and sets. Which leads me to my second question…
2) When simpler proofs do exist, is it tradition in mathematics to honor the original proofs as a matter of principle, publishing them instead of more streamlined versions?
Proof that $\mathbb{R}^J$ is not normal:
The proof begins by examining a closed subspace of $\mathbb{R}^J$, in this case $X = (\mathbb{Z}_+)^J$, and showing that it is not normal. The elements in $X$ are written using the map notation, $x:J \rightarrow \mathbb{Z}_+$.
First, in part (a), it's shown that if $x \in X$ and $B$ is a finite subset of $J$, then the sets $U(x, B) = \{ y \in X : y(\alpha) = x(\alpha), \alpha \in B\}$ form a basis for $X$.
Second, in part (b), it's shown that if $P_n$ is the subset of $X$ consisting of $x \in X$ s.t. $x$ is injective on $J – x^{-1}(n)$, that $P_n$ is closed. Moreover, if $n \neq m$, then $P_n \cap P_m = \emptyset$. The proof of this utilizes the basis elements in (a).
Here's where my proof differs from that of the literature. Consider a set $P_n$ for some $n \in \mathbb{Z}^+$, and suppose that $U$ is an open set containing $P_n$. We will show that $U = X$, and so $X$ cannot be normal, since two closed sets $P_n$ and $P_m$ cannot be separated by disjoint open sets.
Choose some $\alpha' \in J$, and define a set of sequences $x_i \in X$ as follows, where $i \in \mathbb{Z}^+$: Let $x_i(\alpha) = i$ when $\alpha = \alpha'$, and $x_i(\alpha) = n$ otherwise. Then each sequence $x_i \in P_n$, since $x_i(\alpha) = n$ on all but at most a singleton in $J$, namely $\alpha'$. Thus, $x_i \in U$ for each $i \in \mathbb{Z}^+$. As such, the coordinate corresponding to $\alpha'$ in the product of open sets that make up $U$ must be equal to $\mathbb{Z}^+$. Since $\alpha'$ was arbitrary, every coordinate in the product $U$ must be equal to $\mathbb{Z}^+$, and so $U = X$.
Thus $X$ cannot be normal, as two disjoint closed sets, $P_n$ and $P_m$ where $n \neq m$, cannot be separated by disjoint open sets.
Best Answer
Even without seeing just where the problem is, you can see that there must be one: if $m \ne n$, then $X\setminus P_m$ is an open nbhd of $P_n$ properly contained in $X$, so your conclusion that there can be no such set is false.
The flaw in your argument is that the open nbhd $U$ of $P_n$ need not be a basic open nbhd: it need not be a product of open sets. In fact, no set of the form $U(x,B)$ can contain $P_n$. To see this, let $B_0 = \{\alpha\in B:x(\alpha)=n\}$, and let $B_1 = B \setminus B_0$. If $B_1 \ne \varnothing$, define $y \in X$ by $y(\alpha) = n$ for all $\alpha \in J$; clearly $y \in P_n \setminus U(x,B)$. If $B_1 = \varnothing$, fix $\alpha_0 \in B_0$, and define $y \in X$ by $$y(\alpha) = \begin{cases}n+1,&\alpha = \alpha_0\\ n,&\text{otherwise}; \end{cases}$$ clearly we again have $y \in P_n \setminus U(x,B)$.