Kojman-Shelah scale subspace

Assume AC. Take the infinite $B\subseteq \omega \setminus \{0,1\}$ and the Rudin space $X_R(B)$ of Rudin ordinal box spaces on infinite index sets. Set $\lambda ={\mathrm{\aleph }}_{\omega +1}$, and fix a normalized scale $(f_{\alpha }{)}_{\alpha <\lambda }$ in ${\prod }_{n\in B}{\mathrm{\aleph }}_n$ as in Normalizing a scale at existing least upper bounds. Thus $f_{\alpha }{<}^{\ast }{f}_{\beta }$ for $\alpha <\beta$, the sequence is cofinal in the eventual order, and at each limit $\delta <\lambda$ of uncountable cofinality where the preceding sequence has a least upper bound modulo finite, $f_{\delta }$ is such a least upper bound. Existence of a least bound at every limit is not assumed.

The Kojman–Shelah scale subspace is

$X=\{h\in X_R(B):(\mathrm{\exists }\alpha <\lambda )h{=}^{\ast }{f}_{\alpha }\},$

with its subspace topology. Here $h{=}^{\ast }f$ means that $\{n\in B:h(n)\mathrm{\ne }f(n)\}$ is finite, and $h{<}^{\ast }f$ means that $\{n\in B:h(n)\ge f(n)\}$ is finite. Pointwise comparison retains its separate meaning at every coordinate.

Only eventual-equivalence classes which meet $X_R(B)$ contribute points to $X$. In particular this definition does not assert that any given $f_{\alpha }$ belongs to $X_R(B)$, or that every class is nonempty. An admissible finite modification means changing finitely many coordinates while keeping the resulting point in $X_R(B)$; such a modification preserves membership in the displayed set $X$, since the union of its modification set with the original finite exceptional set is finite.