Realcompactification of Rudin's Dowker space

Question

Let $$X = \{f\in \prod_n (\omega_n+1) : \exists_{i} \forall_n\omega_0 < \text{cf}(f(n)) < \omega_i\}$$ be Rudin's Dowker space and $$X' = \{f\in \prod_n (\omega_n+1) : \forall_n \omega_0 < \text{cf}(f(n))\}.$$

In the original paper by Rudin, namely A normal space $X$ for which $X\times I$ is not normal, its claimed that $X'$ is the Hewitt realcompactification of $X$. Part of the proof is that every continuous function $g:X\to \mathbb{R}$ extends to $\tilde{g}:X'\to\mathbb{R}$. This is supposed to follow from lemma $5$ of that paper. I would like someone to help me prove it.

Lemma 5. Let $\mathcal{H}$ be a discrete family of closed subsets of $X$, $U\subseteq X$ open, $t\in \prod_n (\omega_n+1)$ be defined as $t(n) = \sup\{h(n) : h\in U\}$, $\text{cf}(t(n)) > \omega_0$ for all $n$, then there is $f\in \prod_n (\omega_n+1)$ such that $f < t$ and $U\cap \{h : f < h\}$ intersects at most one element of $\mathcal{H}$.

Answer 1

Let $F = \prod_n (\omega_n+1)$. Pick $p\in X'\setminus X$ and take $U = \{h\in X : h \leq p\}$. Then $t = p$.

Take $\mathcal{H}_n = \{g^{-1}([\frac{2m}{n}, \frac{2m+1}{n}]) : m\in\mathbb{N}\}$ and $\mathcal{K}_n = \{g^{-1}([\frac{2m-1}{n}, \frac{2m}{n}]) : m\in\mathbb{N}\}$ for $n\in\mathbb{N}$. Its easy to see that those are discrete families of closed subsets.

By lemma $5$ there exist $f_n, f_n'\in F$ with $f_n, f_n' < p$ and $\{h\in X : f_n < h\leq p\}$, $\{h\in X : f_n' < h\leq p\}$ intersect at most one element of $\mathcal{H}_n, \mathcal{K}_n$. If they intersect disjoint sets $A\in\mathcal{H}_n, B\in\mathcal{K}_n$, then we can further take $h_n\in F$, $h_n < p$ such that $\{h\in X : h_n < h\leq p\}$ intersects at most one $A, B$ and $f_n, f_n'\leq h_n$. So $\{h\in X : h_n < h\leq p\}$ intersects at most one element of $\{g^{-1}([\frac{m}{n}, \frac{m+1}{n}]) : m\in\mathbb{Z}\}$. And if $A, B$ intersect then we can take $h_n\in F$, $h_n(i) = \max(f_n(i), f_n'(i))$, $h_n < p$ such that $\{h\in X : h_n < h\leq p\}$ is contained in some element of $\{g^{-1}([\frac{m}{n}, \frac{m+2}{n}]) : m\in\mathbb{Z}\}$.

Either way we see from this that one can find $f\in F$ with $f < p$ such that $\{h\in X : f < h\leq p\}\subseteq g^{-1}(a)$ for some $a\in\mathbb{R}$, just take $f(i) = \sup_n h_n(i)$. Let $g'(p) = a$ and $g'(h) = g(h)$ for $h\in X$. Then $g'$ is continuous on the clopen set $\{h\in X\cup \{p\} : f < h\leq p\}$ since its constant there. So $g':X\cup \{p\}\to\mathbb{R}$ is continuous.

Since $g$ can be extended to $X\cup \{p\}$ for all $p\in X'$ and $X$ is dense in $X'$, $g$ can be extended to all of $X'$.