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# Models in Which Every Nonmeager Set is Nonmeager in a Nowhere Dense Cantor Set

Published:2005-12-01
Printed: Dec 2005
• Maxim R. Burke
• Arnold W. Miller
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## Abstract

We prove that it is relatively consistent with $\ZFC$ that in any perfect Polish space, for every nonmeager set $A$ there exists a nowhere dense Cantor set $C$ such that $A\cap C$ is nonmeager in $C$. We also examine variants of this result and establish a measure theoretic analog.
 Keywords: Property of Baire, Lebesgue measure, Cantor set, oracle forcing
 MSC Classifications: 03E35 - Consistency and independence results 03E17 - Cardinal characteristics of the continuum 03E50 - Continuum hypothesis and Martin's axiom [See also 03E57]

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