In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a cardinal function (or cardinal invariant) is a
function that returns
cardinal number
In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the cas ...
s.
Cardinal functions in set theory
* The most frequently used cardinal function is the function that assigns to a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
''A'' its
cardinality
The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
, denoted by , ''A'', .
*
Aleph number
In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. They were introduced by the mathematician Georg Cantor and are named after the symbol he used t ...
s and
beth number
In mathematics, particularly in set theory, the beth numbers are a certain sequence of infinite cardinal numbers (also known as transfinite numbers), conventionally written \beth_0, \beth_1, \beth_2, \beth_3, \dots, where \beth is the Hebrew lett ...
s can both be seen as cardinal functions defined on
ordinal number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the leas ...
s.
*
Cardinal arithmetic
In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the case ...
operations are examples of functions from cardinal numbers (or pairs of them) to cardinal numbers.
* Cardinal characteristics of a (proper)
ideal ''I'' of
subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
s of ''X'' are:
:
::The "additivity" of ''I'' is the smallest number of sets from ''I'' whose
union is not in ''I'' any more. As any ideal is closed under finite unions, this number is always at least
; if ''I'' is a σ-ideal, then
:
:: The "covering number" of ''I'' is the smallest number of sets from ''I'' whose union is all of ''X''. As ''X'' itself is not in ''I'', we must have add(''I'') ≤ cov(''I'').
:
:: The "uniformity number" of ''I'' (sometimes also written
) is the size of the smallest set not in ''I''. Assuming ''I'' contains all
singletons, add(''I'') ≤ non(''I'').
:
:: The "cofinality" of ''I'' is the
cofinality of the
partial order
In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements needs to be comparable ...
(''I'', ⊆). It is easy to see that we must have non(''I'') ≤ cof(''I'') and cov(''I'') ≤ cof(''I'').
:In the case that
is an ideal closely related to the structure of the
reals, such as the ideal of
Lebesgue null sets or the ideal of
meagre sets, these cardinal invariants are referred to as
cardinal characteristics of the continuum.
* For a
preordered set the bounding number
and dominating number
are defined as
::
::
* In
PCF theory the cardinal function
is used.
Cardinal functions in topology
Cardinal functions are widely used in
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
as a tool for describing various
topological properties
In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological space ...
. Below are some examples. (Note: some authors, arguing that "there are no finite cardinal numbers in
general topology
In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differ ...
", prefer to define the cardinal functions listed below so that they never taken on finite cardinal numbers as values; this requires modifying some of the definitions given below, for example by adding "
" to the right-hand side of the definitions, etc.)
* Perhaps the simplest cardinal invariants of a
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
are its cardinality and the cardinality of its topology, denoted respectively by
and
* The
weight
In science and engineering, the weight of an object is a quantity associated with the gravitational force exerted on the object by other objects in its environment, although there is some variation and debate as to the exact definition.
Some sta ...
of a topological space
is the cardinality of the smallest
base for
When
the space
is said to be ''
second countable''.
** The
-weight of a space
is the cardinality of the smallest
-base for
(A
-base is a set of non-
empty open set
In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line.
In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
s whose supersets includes all opens.)
** The network weight
of
is the smallest cardinality of a network for
A ''network'' is a
family
Family (from ) is a Social group, group of people related either by consanguinity (by recognized birth) or Affinity (law), affinity (by marriage or other relationship). It forms the basis for social order. Ideally, families offer predictabili ...
of sets, for which, for all points
and
open neighbourhoods
containing
there exists
in
for which
* The
character of a topological space
at a point
is the cardinality of the smallest
local base for
The character of space
is
When
the space
is said to be ''
first countable''.
* The density
of a space
is the cardinality of the smallest
dense subset
In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ra ...
of
When
the space
is said to be ''
separable''.
* The
Lindelöf number of a space
is the smallest infinite cardinality such that every
open cover
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a family of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\su ...
has a subcover of cardinality no more than
When
the space
is said to be a ''
Lindelöf space
In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of ''compactness'', which requires the existence of a ''finite'' sub ...
''.
* The cellularity or Suslin number of a space
is
::
:* The hereditary cellularity (sometimes called spread) is the least upper bound of cellularities of its subsets:
or
where "discrete" means that it is a
discrete topological space.
* The extent of a space
is
So
has countable extent exactly when it has no
uncountable
In mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger tha ...
closed discrete subset.
* The tightness
of a topological space
at a point
is the smallest cardinal number
such that, whenever
for some subset
of
there exists a subset
of
with
such that
Symbolically,
The tightness of a space
is
When
the space
is said to be ''
countably generated'' or ''
countably tight''.
** The augmented tightness of a space
is the smallest
regular cardinal such that for any
there is a subset
of
with cardinality less than
such that
Basic inequalities
Cardinal functions in Boolean algebras
Cardinal functions are often used in the study of
Boolean algebras.
[Monk, J. Donald: ''Cardinal invariants on Boolean algebras''. "Progress in Mathematics", 142. Birkhäuser Verlag, Basel, .] We can mention, for example, the following functions:
*Cellularity
of a Boolean algebra
is the supremum of the cardinalities of
antichains in
.
*Length
of a Boolean algebra
is
::
*Depth
of a Boolean algebra
is
::
.
*Incomparability
of a Boolean algebra
is
::
.
*Pseudo-weight
of a Boolean algebra
is
::
Cardinal functions in algebra
Examples of cardinal functions in
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
are:
*
Index of a subgroup
In mathematics, specifically group theory, the index of a subgroup ''H'' in a group ''G'' is the
number of left Coset, cosets of ''H'' in ''G'', or equivalently, the number of right cosets of ''H'' in ''G''.
The index is denoted , G:H, or :Ho ...
''H'' of ''G'' is the number of
coset
In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
s.
*
Dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
of a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
''V'' over a
field ''K'' is the cardinality of any
Hamel basis
In mathematics, a set of elements of a vector space is called a basis (: bases) if every element of can be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as ...
of ''V''.
*More generally, for a
free module
In mathematics, a free module is a module that has a ''basis'', that is, a generating set that is linearly independent. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commu ...
''M'' over a
ring ''R'' we define rank
as the cardinality of any basis of this
module.
*For a
linear subspace
In mathematics, the term ''linear'' is used in two distinct senses for two different properties:
* linearity of a ''function (mathematics), function'' (or ''mapping (mathematics), mapping'');
* linearity of a ''polynomial''.
An example of a li ...
''W'' of a vector space ''V'' we define
codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties.
For affine and projective algebraic varieties, the codimension equals ...
of ''W'' (with respect to ''V'').
*For any
algebraic structure
In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplicatio ...
it is possible to consider the minimal cardinality of
generators of the structure.
*For
algebraic field extension
In mathematics, particularly in algebra, a field extension is a pair of fields K \subseteq L, such that the operations of ''K'' are those of ''L'' restricted to ''K''. In this case, ''L'' is an extension field of ''K'' and ''K'' is a subfield of ...
s,
algebraic degree and
separable degree
In field theory, a branch of algebra, an algebraic field extension E/F is called a separable extension if for every \alpha\in E, the minimal polynomial of \alpha over is a separable polynomial (i.e., its formal derivative is not the zero polyno ...
are often employed (the algebraic degree equals the dimension of the extension as a vector space over the smaller field).
*For non-algebraic field extensions,
transcendence degree
In mathematics, a transcendental extension L/K is a field extension such that there exists an element in the field L that is transcendental over the field K; that is, an element that is not a root of any univariate polynomial with coefficients ...
is likewise used.
External links
* A Glossary of Definitions from General Topolog
See also
*
Cichoń's diagram
References
*
{{DEFAULTSORT:Cardinal Function
Function
Types of functions