Real Analysis Exchange

Porosity, nowhere dense sets and a theorem of Denjoy

Dave L. Renfro

Abstract

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

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

Dates
First available in Project Euclid: 14 June 2012

https://projecteuclid.org/euclid.rae/1339694085

Mathematical Reviews number (MathSciNet)
MR1407269

Zentralblatt MATH identifier
0883.26002

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

Citation

Renfro, Dave L. Porosity, nowhere dense sets and a theorem of Denjoy. Real Anal. Exchange 21 (1995), no. 2, 572--581. https://projecteuclid.org/euclid.rae/1339694085