Infinite-Dimensionality modulo Absolute Borel Classes

Vitalij Chatyrko¹ ; Yasunao Hattori²

¹ Department of Mathematics Linköping University 581 83 Linköping, Sweden
² Department of Mathematics Shimane University Matsue, 690-8504 Japan

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 56 (2008) no. 2, pp. 163-176.

Résumé

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.

Détail

DOI : 10.4064/ba56-2-7

Keywords: each ordinal leq alpha omega present separable metrizable spaces alpha alpha alpha alpha alpha alpha omega where either trdef cal mbox trsur mathop alpha mbox trind alpha infty mathop alpha mbox trind alpha iii mathop alpha mbox trind alpha mathop alpha mbox trind alpha infty mathop alpha mbox trind alpha mathop alpha mbox trind alpha infty alpha cap mathop alpha mbox trind alpha there exists separable metrizable space alpha alpha mbox trind alpha infty mathop alpha mbox trind alpha infty alpha cap mathop alpha mbox trind alpha infty where alpha resp alpha absolutely additive resp multiplicative borel class