Real Analysis Exchange

Porosity, nowhere dense sets and a theorem of Denjoy

Dave L. Renfro

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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 \(\mathbb{R}\).

Article information

Real Anal. Exchange, Volume 21, Number 2 (1995), 572-581.

First available in Project Euclid: 14 June 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 28A99: None of the above, but in this section
Secondary: 54E52: Baire category, Baire spaces

porous set nowhere dense set density in Cantor sets


Renfro, Dave L. Porosity, nowhere dense sets and a theorem of Denjoy. Real Anal. Exchange 21 (1995), no. 2, 572--581.

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