In
mathematics, a porous set is a concept in the study of
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general sett ...
s. Like the concepts of
meagre and
measure zero
In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length.
The notion of null ...
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
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies ...
metric space and let ''E'' be a subset of ''X''. Let ''B''(''x'', ''r'') denote the
closed ball
In mathematics, a ball is the solid figure bounded by a '' sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them).
These concepts are defi ...
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 ''r''
0 > 0 such that, for every 0 < ''r'' ≤ ''r''
0 and every ''x'' ∈ ''X'', there is some point ''y'' ∈ ''X'' with
:
A subset of ''X'' is called ''σ''-porous if it is a
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...
union
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
of porous subsets of ''X''.
Properties
* Any porous set is
nowhere dense
In mathematics, a subset of a topological space is called nowhere dense or rare if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the topology on the space) anywhere. ...
. Hence, all ''σ''-porous sets are meagre sets (or of the
first category
In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is called ...
).
* If ''X'' is a finite-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
R
''n'', then porous subsets are sets of
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides ...
zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usu ...
.
* However, there does exist a non-''σ''-porous subset ''P'' of R
''n'' 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''.
References
*
* {{MathSciNet, id=943561
Metric geometry