Transfinite large inductive dimensions modulo absolute Borel classes

摘要

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:rn(ⅰ) A(0)-trInd ≥ M(0)-trInd ≥ max{A(1)-trInd,M(1)-trInd}, and rn(ⅱ) min{A(α)-trInd, M(α)-trInd} ≥ max{A(β)-trInd, M(β)-trInd}, where 1 ≤ α < β < ω_1.rnWe show that for any two functions a and m from the set of ordinals Ω = {α : α < ω_1} to the set {-1} ∪Ω∪{∞} such thatrn(ⅰ) a(0) ≥ m(0) ≥ max{a(1),m(1)}, andrn(ⅱ) min{a(α),m(α)} ≥ max{a(β), m(β)}, whenever 1 ≤ a < β < ω_1,rnthere is a separable metrizable space X such that A(α)-trInd X = a(α) and M(α)-trInd X = m(α) for each 0 ≤ α < ω_1.