Journal of the Mathematical Society of Japan

Transfinite large inductive dimensions modulo absolute Borel classes

Vitalij A. CHATYRKO and Yasunao HATTORI

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Abstract

The following inequalities between transfinite large inductive dimensions modulo absolutely additive (resp. multiplicative) Borel classes $A(\alpha)$ (resp. $M(\alpha))$ = \begin{equation*} (i) \; \; \; A(0) \mathsf{-trInd} \geq M(0) \mathsf{-trInd} \geq max\{A(1) \mathsf{-trInd}, M(1) \mathsf{-trInd}\}, and \\ (ii) \; \; \; min \{A(\alpha) \mathsf{-trInd},M(\alpha) \mathsf{-trInd}\} \geq max \{A(\beta) \mathsf{-trInd},M(\beta) \mathsf{-trInd}\}, \\ \mathsf{where \;} 1 \leq \alpha "< \beta < \omega_1. \end{equation*} We show that for any two functions $a$ and $m$ from the set of ordinals $\Omega = \{ \alpha : \alpha < \omega_1 \}$ to the set $\{ -1 \} \cup \Omega \cup \{ \infty \}$ such that

\begin{equation*} (i) \; \; \; a(0) \geq m(0) \geq max\{a(1),m(1)\}, \mathsf{and}\\ (ii) \; \; \; min \{a(\alpha),m(\alpha) \} \geq max \{ a(\beta),m(\beta)\}, \mathsf{whenever \;} 1 \geq \alpha < \beta < \omega_1, \end{equation*} there is a separable metrizable space X such that $A(\alpha) \mathsf{-trInd}X = a(/alpha)$ and $M(\alpha) \mathsf{-trInd}X = m(\alpha)$ for each $0 \geq \alpha < \omega_1$.

Article information

Source
J. Math. Soc. Japan, Volume 61, Number 2 (2009), 327-344.

Dates
First available in Project Euclid: 13 May 2009

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1242220712

Digital Object Identifier
doi:10.2969/jmsj/06120327

Mathematical Reviews number (MathSciNet)
MR2532891

Zentralblatt MATH identifier
1173.54014

Subjects
Primary: 54F45: Dimension theory [See also 55M10]
Secondary: 04A15 54D35: Extensions of spaces (compactifications, supercompactifications, completions, etc.) 54H05: Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) [See also 03E15, 26A21, 28A05]

Keywords
inductive dimensions modulo $mathcal{P}$ absolute Borel class absolutely multipricative Borel class absolutely additive Borel class separable metrizable space

Citation

CHATYRKO, Vitalij A.; HATTORI, Yasunao. Transfinite large inductive dimensions modulo absolute Borel classes. J. Math. Soc. Japan 61 (2009), no. 2, 327--344. doi:10.2969/jmsj/06120327. https://projecteuclid.org/euclid.jmsj/1242220712


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