On dimensional properties of the generalized Smirnov's spaces

We show that the transfinite inductive dimensions modulo P P-trind and P-trInd introduced in M.G. Charalambous (1997) [2] differ by simple spaces, where P is the absolutely additive Borel class A(α) or the absolutely multiplicative Borel class M(α), 0⩽α<ω1.