Infinite-Dimensionality modulo Absolute Borel Classes
Tom 56 / 2008
Bulletin Polish Acad. Sci. Math. 56 (2008), 163-176
MSC: Primary 54F45; Secondary 04A15, 54D35, 54H05.
DOI: 10.4064/ba56-2-7
Streszczenie
For each ordinal $1 \leq \alpha < \omega_1$ 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.
