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Silver Measurability and its relation to other regularity properties

Published online by Cambridge University Press:  03 February 2005

JÖRG BRENDLE
Affiliation:
The Graduate School of Science and Technology, Kobe University, Rokko-dai 1-1, Nada, Kobe 657-8501, Japan. e-mail: brendle@kurt.scitec.kobe-u.ac.jp
LORENZ HALBEISEN
Affiliation:
Department of Pure Mathematics, Queen's University Belfast, Belfast BT7 1NN, Northern Ireland. e-mail: halbeis@qub.ac.uk
BENEDIKT LÖWE
Affiliation:
Institute for Logic, Language and Computation, Universiteit van Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands. e-mail: bloewe@science.uva.nl

Abstract

For $a\subs b\subs\omega$ with $b{\setminus} a$ infinite, the set $D\,{=}\,\{x\in\reals\,{:}\, a\subs x\subs b\}$ is called a doughnut. Doughnuts are equivalent to conditions of Silver forcing, and so, a set $S\subs\reals$ is called Silver measurable, or completely doughnut, if for every doughnut $D$ there is a doughnut $D'\subs D$ which is contained in or disjoint from $S$. In this paper, we investigate the Silver measurability of $\bDelta$ and $\bSigma$ sets of reals and compare it to other regularity properties like the Baire and the Ramsey property and Miller and Sacks measurability.

Type
Research Article
Copyright
2005 Cambridge Philosophical Society

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