Porous set

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 r₀ > 0 such that, for every 0 < r ≤ r₀ and every x ∈ X, there is some point y ∈ X with

B ( y , α r ) ⊆ B ( x , r ) ∖ 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 > 0, there is a point y ∈ X and s > 0 such that

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.