In mathematics, a cardinal function (or cardinal invariant) is a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
that returns
cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. Th ...
s.
Cardinal functions in set theory
* The most frequently used cardinal function is a 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
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
, 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 that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named af ...
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 second H ...
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 least n ...
s.
*
Cardinal arithmetic
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The ...
operations are examples of functions from cardinal numbers (or pairs of them) to cardinal numbers.
* Cardinal characteristics of a (proper)
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
''I'' of subsets 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
In mathematics, especially in order theory, the cofinality cf(''A'') of a partially ordered set ''A'' is the least of the cardinalities of the cofinal subsets of ''A''.
This definition of cofinality relies on the axiom of choice, as it uses the ...
of the
partial order
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
(''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 set
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 calle ...
s, these cardinal invariants are referred to as
cardinal characteristics of the continuum.
* For a
preordered set
In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive relation, reflexive and Transitive relation, transitive. Preorders are more general than equivalence relations and (non-strict) partia ...
the bounding number
and dominating number
are defined as
::
::
* In
PCF theory PCF theory is the name of a mathematical theory, introduced by Saharon , that deals with the cofinality of the ultraproducts of ordered sets. It gives strong upper bounds on the cardinalities of power sets of singular cardinals, and has many more ap ...
the cardinal function
is used.
Cardinal functions in topology
Cardinal functions are widely used in
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
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 spac ...
. Below are some examples. (Note: some authors, arguing that "there are no finite cardinal numbers in
general topology
In mathematics, general 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 differential topology, geomet ...
", 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
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 the force acting on the object due to gravity.
Some standard textbooks define weight as a Euclidean vector, vector quantity, the gravitational force acting on the object. Others define weigh ...
of a topological space
is the cardinality of the smallest
base for
When
the space
is said to be ''
second countable
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...
''.
** The
-weight of a space
is the cardinality of the smallest
-base for
(A
-base is a set of nonempty opens whose supersets includes all opens.)
** The network weight
of
is the smallest cardinality of a network for
A ''network'' is a family
of sets, for which, for all points
and open neighbourhoods
containing
there exists
in
for which
* The
character
Character or Characters may refer to:
Arts, entertainment, and media Literature
* ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk
* ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
of a topological space
at a point
is the cardinality of the smallest
local base In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x.
Definitions
Neighbour ...
for
The character of space
is
When
the space
is said to be ''
first countable
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base) ...
''.
* 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 collection 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\s ...
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
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
.
* The extent of a space
is
So
has countable extent exactly when it has no uncountable 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
In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. More explicitly, this means that \kappa is a regular cardinal if and only if every unbounded subset C \subseteq \kappa has cardinality \kappa. Infinite ...
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
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a gen ...
.
[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
antichain
In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two distinct elements in the subset are incomparable.
The size of the largest antichain in a partially ordered set is known as its wid ...
s 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 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 cosets of ''H'' in ''G'', or equivalently, the number of right cosets of ''H'' in ''G''.
The index is denoted , G:H, or :H/math> or (G ...
''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 Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
of a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
''V'' over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
''K'' is the cardinality of any
Hamel basis
In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as component ...
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 consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in t ...
''M'' over a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
''R'' we define rank
as the cardinality of any basis of this module.
*For a
linear subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, 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 the ...
of ''W'' (with respect to ''V'').
*For any
algebraic structure
In mathematics, an algebraic structure 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 multiplication), and a finite set of ...
it is possible to consider the minimal cardinality of
generator
Generator may refer to:
* Signal generator, electronic devices that generate repeating or non-repeating electronic signals
* Electric generator, a device that converts mechanical energy to electrical energy.
* Generator (circuit theory), an eleme ...
s of the structure.
*For
algebraic extension
In mathematics, an algebraic extension is a field extension such that every element of the larger field is algebraic over the smaller field ; that is, if every element of is a root of a non-zero polynomial with coefficients in . A field ext ...
s,
algebraic degree
Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings.
Algebraic may also refer to:
* Algebraic data type, a data ...
and
separable degree are often employed (note that the algebraic degree equals the dimension of the extension as a vector space over the smaller field).
*For non-algebraic
field extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
s,
transcendence degree
In abstract algebra, the transcendence degree of a field extension ''L'' / ''K'' is a certain rather coarse measure of the "size" of the extension. Specifically, it is defined as the largest cardinality of an algebraically independent subset of ...
is likewise used.
External links
* A Glossary of Definitions from General Topolog
See also
*
Cichoń's diagram In set theory,
Cichoń's diagram or Cichon's diagram is a table of 10 infinite cardinal numbers related to the set theory of the reals displaying the provable relations between these
Cardinal characteristic of the continuum, cardinal characterist ...
References
*
{{DEFAULTSORT:Cardinal Function
Function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
Types of functions