The ambient Rudin box space

In the AC setting, with $B$, $P_B$ and box topology of Rudin ordinal box spaces on infinite index sets, let $t\in P_B$ be the coordinate top $t(n)={\mathrm{\aleph }}_n$. Define the ambient Rudin space

$Y_B=\{h\in P_B:\mathrm{cf}(h(n))>\omega \text{ for every }n\in B\},$

with the topology induced from $P_B$. The Rudin space $X_R(B)$ is its subspace obtained by imposing a single finite-aleph bound on all coordinate cofinalities. No such uniform bound is part of the definition of $Y_B$.

For $a,b\in P_B$ with $a<b$ pointwise, set

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

The upper endpoint $b$ is an ordinal function in $P_B$ and is not required to be a point of $Y_B$; this distinction matters when partitions lower a coordinate endpoint to an ordinal of countable cofinality. Every coordinate of a point of $Y_B$ is a nonzero limit ordinal, since zero and successor ordinals have cofinality zero and one, respectively.