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1995/1996 Porosity, nowhere dense sets and a theorem of Denjoy
Dave L. Renfro
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Real Anal. Exchange 21(2): 572-581 (1995/1996).


In the 1940’s, A. Denjoy proved that the typical point of a perfect nowhere dense set in \(\mathbb{R}\) is a point of strong porosity for that set. We prove two stronger versions of this for arbitrary metric spaces. Theorem 3 says that if \(E\) is any closed nowhere dense set in a metric space, and \(h\) is any porosity scale function, then the typical point in \(E\) is a point at which \(E\) is \(h\)-porous. Thus, if the metric space is complete, then “most” points of \(E\) are points at which \(E\) is very thin in the sense of porosity. Theorem 4 says that if \(F\) is closed and \(h\)-porous in \(\mathbb{R}\), then there exists a closed nowhere dense set \(E\) in \(\mathbb{R}\) containing \(F\) such that \(F\) is \(h\)-porous in the subspace \(E\). Therefore, in the sense of \(h\)-porosity, no nontrivial information about the porosity of a closed set in \(\mathbb{R}\) can be inferred from its porosity relative to some closed nowhere dense set in \(\mathbb{R}\).


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Dave L. Renfro. "Porosity, nowhere dense sets and a theorem of Denjoy." Real Anal. Exchange 21 (2) 572 - 581, 1995/1996.


Published: 1995/1996
First available in Project Euclid: 14 June 2012

zbMATH: 0883.26002
MathSciNet: MR1407269

Primary: 28A99
Secondary: 54E52

Keywords: density in Cantor sets , nowhere dense set , porous set

Rights: Copyright © 1995 Michigan State University Press

Vol.21 • No. 2 • 1995/1996
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