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2002/2003 Small opaque sets.
Ondřej Zindulka
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Real Anal. Exchange 28(2): 455-470 (2002/2003).


A set in a separable metric space is called \emph{Borel--opaque} if it meets every Borel set of positive topological dimension. We show that if there is a set of reals with cardinality of the continuum and universal measure zero, then each separable space contains a Borel--opaque set that is of universal measure zero. Similar results hold for opaque sets that are perfectly meager, \la sets, \lap sets etc., and can be extended to some nonseparable spaces. On the other hand, we show that a \s set is zero--dimensional. Using opacity we also construct universal measure zero sets of positive Hausdorff dimension.


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Ondřej Zindulka. "Small opaque sets.." Real Anal. Exchange 28 (2) 455 - 470, 2002/2003.


Published: 2002/2003
First available in Project Euclid: 20 July 2007

zbMATH: 1044.28011
MathSciNet: MR2009767

Primary: 28A78 , 28C15 , 54F45

Keywords: \la set , \lap set , \s set , opaque set , perfectly meager , topological dimension , universal measure zero , universally meager , zero--dimensional

Rights: Copyright © 2002 Michigan State University Press

Vol.28 • No. 2 • 2002/2003
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