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2008/2009 On ℓp-like Equivalence Relations
Tamás Mátrai
Real Anal. Exchange 34(2): 377-412 (2008/2009).


For $f : [0,1] \to \mathbb{R}^{+}$, consider the relation $\mathbf{E}_{f}$ on $[0,1]^{\omega}$ defined by $(x_{n}) \mathbf{E}_{f} (y_{n}) \Leftrightarrow \sum_{n \omega} f(|y_{n} - x_{n}|) \infty.$ We study the Borel reducibility of Borel equivalence relations of the form $\mathbf{E}_{f}$. Our results indicate that for every $1 \leq p q \infty$, the order $\leq_{B}$ of Borel reducibility on the set of equivalence relations $\mathbf{E}:\mathbf{E}_{Id^\mathcal{p}} \leq_B \mathbf{E} \leq_B \mathbf{E}_{Id^\mathcal{q}}$ is more complicated than expected, e.g.\ consistently every linear order of cardinality continuum embeds into it


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Tamás Mátrai. "On ℓp-like Equivalence Relations." Real Anal. Exchange 34 (2) 377 - 412, 2008/2009.


Published: 2008/2009
First available in Project Euclid: 29 October 2009

MathSciNet: MR2569194

Primary: 03E15
Secondary: 46A45

Keywords: $\ell^p$ space , $E_1$ , Borel equivalence relation , Borel reduction , Lipschitz embedding

Rights: Copyright © 2008 Michigan State University Press

Vol.34 • No. 2 • 2008/2009
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