Transfinite large inductive dimensions modulo absolute Borel classes

Abstract

The following inequalities between transfinite large inductive dimensions modulo absolutely additive (resp. multiplicative) Borel classes $A(\alpha)$ (resp. $M(\alpha))$ = \begin{equation*} (i) \; \; \; A(0) \mathsf{-trInd} \geq M(0) \mathsf{-trInd} \geq max\{A(1) \mathsf{-trInd}, M(1) \mathsf{-trInd}\}, and \\ (ii) \; \; \; min \{A(\alpha) \mathsf{-trInd},M(\alpha) \mathsf{-trInd}\} \geq max \{A(\beta) \mathsf{-trInd},M(\beta) \mathsf{-trInd}\}, \\ \mathsf{where \;} 1 \leq \alpha "< \beta < \omega_1. \end{equation*} We show that for any two functions $a$ and $m$ from the set of ordinals $\Omega = \{ \alpha : \alpha < \omega_1 \}$ to the set $\{ -1 \} \cup \Omega \cup \{ \infty \}$ such that

\begin{equation*} (i) \; \; \; a(0) \geq m(0) \geq max\{a(1),m(1)\}, \mathsf{and}\\ (ii) \; \; \; min \{a(\alpha),m(\alpha) \} \geq max \{ a(\beta),m(\beta)\}, \mathsf{whenever \;} 1 \geq \alpha < \beta < \omega_1, \end{equation*} there is a separable metrizable space X such that $A(\alpha) \mathsf{-trInd}X = a(/alpha)$ and $M(\alpha) \mathsf{-trInd}X = m(\alpha)$ for each $0 \geq \alpha < \omega_1$.