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 be a complete [metric space] and let E be a subset of X. Let B denote the closed ball in with centre x ∈ X and radius r > 0. E is said to be porous if there exist constants 0 <; α < 1 and r₀ > 0 such that, for every 0 < r ≤ r₀ and every x ∈ X, there is some point y ∈ 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.
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 > 0, there is a point y ∈ X and s > 0 such that