Rudin ordinal box spaces on infinite index sets

Assume AC. Fix an infinite $B\subseteq \omega \setminus \{0,1\}$ and put

$P_B={\prod }_{n\in B}({\mathrm{\aleph }}_n+1)=\{h:\mathrm{dom}h=B,h(n)\le {\mathrm{\aleph }}_n\text{ for every }n\in B\}\mathrm{.}$

Each factor is the ordinal interval $[0,{\mathrm{\aleph }}_n]$ with its order topology, as in The order topology on an ordinal, with the half-open intervals $(\alpha ,\beta ]$ and the initial segments $[0,\beta ]$ as a basis. Give $P_B$ the box topology: its open sets are unions of boxes ${\prod }_{n\in B}{O}_n$ with every $O_n$ open in its factor. The whole product is such a box, and the intersection of two boxes is the box with factors $O_n\cap V_n$, so this prescription defines a topology. In contrast, The product set ${\prod }_{i\in I}{X}_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space restricts a basic box to have only finitely many factors different from the whole factor. No such restriction is imposed here.

The Rudin space on $B$ is the subspace

$X_R(B)=\{h\in P_B:\text{there is }m<\omega \text{ such that }\omega <\mathrm{cf}(h(n))<{\mathrm{\aleph }}_m\text{ for every }n\in B\}\mathrm{.}$

The witness $m$ may depend on the point $h$, but one $m$ must bound the cofinalities of all its coordinates. Both inequalities are strict. The coordinate endpoint $h(n)={\mathrm{\aleph }}_n$ is allowed when the uniform cofinality condition permits it.

Use pointwise $a<b$ to mean $a(n)<b(n)$ at every coordinate, and define

$(a,b{]}_R=\{h\in X_R(B):a(n)<h(n)\le b(n)\text{ for every }n\in B\}\mathrm{.}$

The bounds $a,b$ are members of $P_B$ with $a<b$; they need not belong to $X_R(B)$. These half-open boxes, and in particular $(a,h{]}_R$ at $h\in X_R(B)$, give the local base proved in Clopen boxes and the P-space property ↗. Eventual comparisons remain those of Reduced products, true cofinality and scales; pointwise and eventual comparisons are distinct conventions throughout.