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, 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 &gt; 0. E is said to be porous if there exist constants 0 &lt; &alpha; &lt; 1 and r0 &gt; 0 such that, for every 0 &lt; r ≤ r0 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 Rn, then porous subsets are sets of Lebesgue measure zero.
 * However, there does exist a non-σ-porous subset P of Rn 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 &gt; 0, there is a point y ∈ X and s &gt; 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 = &alpha;r (at least for small enough r), where 0 &lt; &alpha; &lt; 1 is a constant that depends only on E.