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(α) (resp. M(α)) hold in separable metrizable spaces:
(i) A(0)-trInd≥M(0)-trInd≥max{A(1)-trInd, M(1)-trInd}, and
(ii) min{A(α)-trInd, M(α)-trInd}≥max{A(β)-trInd, M(β)-trInd}, where 1≤α<β<ω₁.
We show that for any two functions a and m from the set of ordinals Ω={α:α<ω₁} to the set {−1}∪Ω∪{∞} such that
(i) a(0)≥m(0)≥max{a(1), m(1)}, and
(ii) min{a(α),m(α)}≥max{a(β),m(β)}, whenever 1≤α<β<ω₁,
there is a separable metrizable space X such that A(α)-trIndX=a(α) and M(α)-trIndX=m(α) for each 0≤α<ω₁.