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Porous set

In mathematics, a porous set is a concept in the study of metric spaces. Like the concepts of meagre and measure zero sets, a porous set can be considered "sparse" or "lacking bulk"; however, porous sets are not equivalent to either meagre sets or measure zero sets, as shown below.

Definition

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

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<sup>n</sup>, then porous subsets are sets of Lebesgue measure zero.
  • However, there does exist a non-σ-porous subset P of R<sup>n</sup> 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&nbsp;∈&nbsp;X and r&nbsp;&gt;&nbsp;0, there is a point y&nbsp;∈&nbsp;X and s&nbsp;&gt;&nbsp;0 such that
:
However, if E is also porous, then it is possible to take s&nbsp;=&nbsp;&alpha;r (at least for small enough r), where 0&nbsp;&lt;&nbsp;&alpha;&nbsp;&lt;&nbsp;1 is a constant that depends only on E.

References