Porous set
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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.
[edit] 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
A subset of X is called σ-porous if it is a countable union of porous subsets of X.
[edit] 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.
