Real Analysis Exchange

Small opaque sets.

Ondřej Zindulka

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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.

Article information

Real Anal. Exchange, Volume 28, Number 2 (2002), 455-470.

First available in Project Euclid: 20 July 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 28C15: Set functions and measures on topological spaces (regularity of measures, etc.) 54F45: Dimension theory [See also 55M10] 28A78: Hausdorff and packing measures

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


Zindulka, Ondřej. Small opaque sets. Real Anal. Exchange 28 (2002), no. 2, 455--470.

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