Infinite-Dimensionality modulo Absolute Borel Classes
Volume 56 / 2008
Bulletin Polish Acad. Sci. Math. 56 (2008), 163-176
MSC: Primary 54F45; Secondary 04A15, 54D35, 54H05.
DOI: 10.4064/ba56-2-7
Abstract
For each ordinal we present separable metrizable spaces X_\alpha, Y_\alpha and Z_\alpha such that
(i) {\rm f}\,X_\alpha, f Y_\alpha, f Z_\alpha = \omega_0, where \rm f is either \rm trdef or {\cal K}_0\mbox{-trsur},
(ii) \mathop{A(\alpha)\mbox{-trind}} X_\alpha = \infty and \mathop{M(\alpha)\mbox{-trind}} X_\alpha = -1,(iii) \mathop{A(\alpha)\mbox{-trind}} Y_\alpha = -1 and \mathop{M(\alpha)\mbox{-trind}} Y_\alpha = \infty, and
(iv) \mathop{A(\alpha)\mbox{-trind}} Z_\alpha = \mathop{M(\alpha)\mbox{-trind}} Z_\alpha = \infty and A(\alpha+1) \cap \mathop{M(\alpha+1)\mbox{-trind}} Z_\alpha = -1.
We also show that there exists no separable metrizable space W_\alpha with A(\alpha)\mbox{-trind}\, W_\alpha \ne \infty, \mathop{M(\alpha)\mbox{-trind}} W_\alpha \ne \infty and A(\alpha) \cap \mathop{M(\alpha)\mbox{-trind}} W_\alpha = \infty, where A(\alpha) (resp. M(\alpha)) is the absolutely additive (resp. multiplicative) Borel class.
