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.
Read more about this topic: Porous Set
Famous quotes containing the word properties:
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—John Locke (1632–1704)
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—Ralph Waldo Emerson (1803–1882)