抄録
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.
