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Infinite-dimensionality modulo absolute Borel classes

Chatyrko V., Hattori Y.

Bulletin of the Polish Academy of Sciences. Mathematics

2008

Vol. 56, no 2

163-176

For each ordinal 1 ≤ α < ω 1 we present separable metrizable spaces Xα, Yα and Zα such that (i) f Xα, f Yα, f Zα = ωo, where f is either trdef or Ko-trsur, (ii) A(α)-trind Xα = ∞ and M(α)-trind Xα = -1, (iii) A(α)-trind Yα = -1 and M(α)-trind Zα = ∞, and (iv) A(α)-trind Zα = M(α)-trind Zα = ∞ and A(α + 1) ∩ M(α + l)-trind Zα = -1. We also show that there exists no separable metrizable space Wa with A(α)-trind Wα ≠ ∞, M(α)-trind Wα ≠ ∞ and A(α) ∩ M(α)-trind Wα = ∞, where A(α) (resp. M(α)) is the absolutely additive (resp. multiplicative) Borel class.