Porous set

Porous set

In mathematics, a porosity is a concept in the study of metric spaces. Like the concepts of meagre and measure zero sets, porosity is a notion of a set being somehow "sparse" or "lacking bulk"; however, porosity is not equivalent to either of the above notions, as shown below.

Definition

Let ("X", "d") be a complete metric space and let "E" be a subset of "X". Let "B"("x", "r") denote the closed ball in ("X", "d") with centre "x" ∈ "X" and radius "r" > 0. "E" is said to be porous if there exist constants 0 < "α" < 1 and "r"0 > 0 such that, for every 0 < "r" ≤ "r"0 and every "x" ∈ "X", there is some point "y" ∈ "X" with

:B(y, alpha r) subseteq B(x, r) setminus E.

A subset of "X" is called "σ"-porous if it is a countable union of porous subsets of "X".

Properties

* Any porous set is nowhere dense. Hence, all "σ"-porous sets are meagre sets (or of the first category).
* If "X" is a finite-dimensional Euclidean space R"n", then porous subsets are sets of Lebesgue measure zero.
* However, there does exist a non-"σ"-porous subset "P" of R"n" which is of the first category and of Lebesgue measure zero. This is known as Zajíček's theorem.
* The relationship between porosity and being nowhere dense can be illustrated as follows: if "E" is nowhere dense, then for "x" ∈ "X" and "r" > 0, there is a point "y" ∈ "X" and "s" > 0 such that

::B(y, s) subseteq B(x, r) setminus E.

: However, if "E" is also porous, then it is possible to take "s" = "αr" (at least for small enough "r"), where 0 < "α" < 1 is a constant that depends only on "E".

References

* cite journal
last = Reich
first = Simeon
coauthors = Zaslavski, Alexander J.
title = Two convergence results for continuous descent methods
journal = Electronic Journal of Differential Equations
volume = 2002
year = 2002
issue = 24
pages = 1–11
issn = 1072-6691
* cite journal
last = Zajíček
first = L.
title = Porosity and "σ"-porosity
journal = Real Anal. Exchange
volume = 13
issue = 2
year = 1987/88
pages = 314–350
issn = 0147-1937 MathSciNet|id=943561


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