Rudin ZFC Dowker space and its size

By F1 and A1, $X$ is Hausdorff and normal. It is $T_1$: for distinct $p,q$, disjoint neighborhoods supplied by Hausdorffness give a neighborhood of $q$ excluding $p$, so $X\setminus \{p\}$ is open as the union of such neighborhoods. If $X$ were countably paracompact, F3 would give open expansions with empty intersection for the decreasing closed sequence in F2. F2 rules out precisely those expansions. Hence $X$ is not countably paracompact, and F4 makes it a Dowker space.

For each $n\in B$ define an injection $e_n:[0,{\mathrm{\aleph }}_n]⟶{\mathrm{\aleph }}_n$ by $e_n({\mathrm{\aleph }}_n)=0$, $e_n(r)=r+1$ for finite $r$, and $e_n(\alpha )=\alpha$ for $\omega \le \alpha <{\mathrm{\aleph }}_n$. The three image parts are respectively zero, the positive finite ordinals, and the infinite ordinals below ${\mathrm{\aleph }}_n$; they are disjoint and the maps on them are injective. Thus $h\mapsto (n\mapsto e_n(h(n)))$ injects the inclusive-top product $P_B$, and hence $X$ by F11, into $Q$. F5, F6 and A1 give $\mathrm{\mid }X\mathrm{\mid }\le {\mathrm{\aleph }}_{\omega }^{{\mathrm{\aleph }}_0}$.

For $n\in B$ and $\alpha <{\mathrm{\aleph }}_n$ set $d_n(\alpha )={\omega }_1\cdot (\alpha +1)$, with ordinal multiplication. By F8 its cardinality is that of ${\omega }_1\times (\alpha +1)$. Since $n\ge 2$, both factors have cardinality at most $\mu =\max ({\mathrm{\aleph }}_1,\mathrm{\mid }\alpha +1\mathrm{\mid })<{\mathrm{\aleph }}_n$. F9 bounds the product size by $\mu$; its ordinal order type is therefore below the initial ordinal ${\mathrm{\aleph }}_n$. These values are strictly increasing with $\alpha$: if $\alpha <\beta$, the first $\alpha +1$ blocks form a proper initial segment of the $\beta +1$ blocks in F8, so their order type is strictly smaller. Each value has cofinality exactly ${\omega }_1$. Its last block has order type ${\omega }_1$ and is cofinal, giving the upper bound. For a set of size less than ${\omega }_1$, its points in that block are bounded there by F7 and F10; a later point in the last block bounds the whole set, since all preceding blocks lie below it. This proves the lower bound on cofinality. At $\alpha =0$ the value is ${\omega }_1$, so the same proof includes zero.

Define $D:Q⟶P_B$ by $D(q)(n)=d_n(q(n))$. Step 1.3 shows that every coordinate is below its factor top and has cofinality ${\omega }_1$. Hence F11 gives $D(q)\in X$, with the strict uniform cofinality bound ${\mathrm{\aleph }}_2$. If $q\mathrm{\ne }{q}^{\mathrm{\prime }}$, they differ at some coordinate $n$; injectivity of $d_n$ then gives $D(q)\mathrm{\ne }D(q^{\mathrm{\prime }})$. Thus $D$ is injective, and F5, F6 and A1 give ${\mathrm{\aleph }}_{\omega }^{{\mathrm{\aleph }}_0}\le \mathrm{\mid }X\mathrm{\mid }$. Combined with step 1.2 this proves equality. Step 1.1 proves the Dowker assertion. All constructions used only ZFC and the stated infinite-coordinate hypotheses. QED.