{{Short description|Function that returns cardinal numbers}} In mathematics, a '''cardinal function''' (or '''cardinal invariant''') is a function that returns cardinal numbers.

== Cardinal functions in set theory ==

* The most frequently used cardinal function is the function that assigns to a set ''A'' its cardinality, denoted by |''A''|. * Aleph numbers and beth numbers can both be seen as cardinal functions defined on ordinal numbers. * Cardinal arithmetic operations are examples of functions from cardinal numbers (or pairs of them) to cardinal numbers. * Cardinal characteristics of a (proper) ideal ''I'' of subsets of ''X'' are: :<math>{\rm add}(I) = \min\{|\mathcal{A}| : \mathcal{A}\subseteq I \wedge \bigcup \mathcal{A} \notin I\}.</math> ::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 <math>\aleph_0</math>; if ''I'' is a &sigma;-ideal, then <math>\operatorname{add}(I) \ge \aleph_1.</math> :<math>\operatorname{cov}(I) = \min\{|\mathcal{A}| : \mathcal{A} \subseteq I \wedge \bigcup \mathcal{A} = X\}.</math> :: 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''{{hairsp}}) ≤ cov(''I''{{hairsp}}). :<math>\operatorname{non}(I) = \min\{|A| : A \subseteq X\ \wedge\ A \notin I\},</math> :: The "uniformity number" of ''I'' (sometimes also written <math>{\rm unif}(I)</math>) is the size of the smallest set not in ''I''. Assuming ''I'' contains all singletons, add(''I''{{hairsp}}) ≤ non(''I''{{hairsp}}). :<math>{\rm cof}(I) = \min\{|\mathcal{B}| : \mathcal{B} \subseteq I \wedge \forall A \in I(\exists B \in \mathcal{B})(A\subseteq B)\}.</math> :: The "cofinality" of ''I'' is the cofinality of the partial order (''I'', &sube;). It is easy to see that we must have non(''I''{{hairsp}}) ≤ cof(''I''{{hairsp}}) and cov(''I''{{hairsp}}) ≤ cof(''I''{{hairsp}}).

:In the case that <math>I</math> 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 <math>(\mathbb{P},\sqsubseteq)</math> the '''bounding number''' <math>{\mathfrak b}(\mathbb{P})</math> and '''dominating number''' <math>{\mathfrak d}(\mathbb{P})</math> are defined as ::<math>{\mathfrak b}(\mathbb{P}) = \min\big\{|Y| : Y \subseteq \mathbb{P}\ \wedge\ (\forall x\in \mathbb{P})(\exists y\in Y)(y\not\sqsubseteq x)\big\},</math> ::<math>{\mathfrak d}(\mathbb{P}) = \min\big\{|Y| : Y \subseteq \mathbb{P}\ \wedge\ (\forall x\in \mathbb{P})(\exists y\in Y)(x\sqsubseteq y)\big\}.</math><!--, where "<math>\exists^\infty n\in\mathbb{N}</math>" means: "there are infinitely many natural numbers ''n'' such that...", and "<math>\forall^\infty n\in\mathbb{N}</math>" means "for all except finitely many natural numbers ''n'' we have...". --> <!-- EDITOR'S NOTE: I have commented out some of the content, because it seemed to refer to material that is no longer part of this page, so that a more knowledgeable editor may decide whether to delete it, or modify the foregoing text to make the commented-out fragment once again relevant in this context.-->

* In PCF theory the cardinal function <math>pp_\kappa(\lambda)</math> is used.<ref>{{cite book |author1=Holz, Michael |author2=Steffens, Karsten |author3=Weitz, Edmund |title=Introduction to Cardinal Arithmetic |publisher=Birkhäuser |year=1999 |isbn=3764361247 |url-access=registration |url=https://archive.org/details/introductiontoca0000holz }}</ref>

== Cardinal functions in topology ==

Cardinal functions are widely used in topology as a tool for describing various topological properties.<ref>{{cite book|last=Juhász|first=István|authorlink=István Juhász (mathematician)|title=Cardinal functions in topology|publisher=Math. Centre Tracts, Amsterdam|year=1979|isbn=90-6196-062-2|url=http://oai.cwi.nl/oai/asset/13055/13055A.pdf|access-date=2012-06-30|archive-date=2014-03-18|archive-url=https://web.archive.org/web/20140318001358/http://oai.cwi.nl/oai/asset/13055/13055A.pdf|url-status=dead}}</ref><ref>{{cite book|last=Juhász|first=István|title=Cardinal functions in topology - ten years later|publisher=Math. Centre Tracts, Amsterdam|year=1980|isbn=90-6196-196-3|url=http://oai.cwi.nl/oai/asset/12982/12982A.pdf|access-date=2012-06-30|archive-date=2014-03-17|archive-url=https://web.archive.org/web/20140317230150/http://oai.cwi.nl/oai/asset/12982/12982A.pdf|url-status=dead}}</ref> Below are some examples. (Note: some authors, arguing that "there are no finite cardinal numbers in general topology",<ref>{{cite book|last=Engelking|first=Ryszard|authorlink=Ryszard Engelking|title=General Topology|publisher=Heldermann Verlag, Berlin|year=1989|isbn=3885380064|edition=Revised|series=Sigma Series in Pure Mathematics|volume=6}}</ref> 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 "<math>\;\; + \;\aleph_0</math>" to the right-hand side of the definitions, etc.)

* Perhaps the simplest cardinal invariants of a topological space <math>X</math> are its cardinality and the cardinality of its topology, denoted respectively by <math>|X|</math> and <math>o(X).</math> * The '''weight''' <math>\operatorname{w}(X)</math> of a topological space <math>X</math> is the cardinality of the smallest base for <math>X.</math> When <math>\operatorname{w}(X) = \aleph_0</math> the space <math>X</math> is said to be ''second countable''. ** The '''<math>\pi</math>-weight''' of a space <math>X</math> is the cardinality of the smallest <math>\pi</math>-base for <math>X.</math> (A <math>\pi</math>-base is a set of non-empty open sets whose supersets includes all opens.) ** The '''network weight''' <math>\operatorname{nw}(X)</math> of <math>X</math> is the smallest cardinality of a network for <math>X.</math> A ''network'' is a family <math>\mathcal{N}</math> of sets, for which, for all points <math>x</math> and open neighbourhoods <math>U</math> containing <math>x,</math> there exists <math>B</math> in <math>\mathcal{N}</math> for which <math>x \in B \subseteq U.</math> * The '''character''' of a topological space <math>X</math> '''at a point''' <math>x</math> is the cardinality of the smallest local base for <math>x.</math> The '''character''' of space <math>X</math> is <math display=block>\chi(X) = \sup \; \{\chi(x,X) : x\in X\}.</math> When <math>\chi(X) = \aleph_0</math> the space <math>X</math> is said to be ''first countable''. * The '''density''' <math>\operatorname{d}(X)</math> of a space <math>X</math> is the cardinality of the smallest dense subset of <math>X.</math> When <math>\rm{d}(X) = \aleph_0</math> the space <math>X</math> is said to be ''separable''. * The '''Lindelöf number''' <math>\operatorname{L}(X)</math> of a space <math>X</math> is the smallest infinite cardinality such that every open cover has a subcover of cardinality no more than <math>\operatorname{L}(X).</math> When <math>\rm{L}(X) = \aleph_0</math> the space <math>X</math> is said to be a ''Lindelöf space''. * The '''cellularity''' or '''Suslin number''' of a space <math>X</math> is :: <math>\operatorname{c}(X) = \sup\{|\mathcal{U}| : \mathcal{U} \text{ is a family of mutually disjoint non-empty open subsets of } X\}.</math> :* The '''hereditary cellularity''' (sometimes called '''spread''') is the least upper bound of cellularities of its subsets: <math display=block>s(X) = {\rm hc}(X) = \sup\{ {\rm c} (Y) : Y \subseteq X \}</math> or <math display=block>s(X) = \sup\{|Y|:Y \subseteq X \text{ with the subspace topology is discrete} \}</math> where "discrete" means that it is a discrete topological space. * The '''extent''' of a space <math>X</math> is <math display=block>e(X) = \sup\{|Y| : Y \subseteq X \text{ is closed and discrete}\}.</math> So <math>X</math> has countable extent exactly when it has no uncountable closed discrete subset. * The '''tightness''' <math>t(x, X)</math> of a topological space <math>X</math> '''at a point''' <math>x \in X</math> is the smallest cardinal number <math>\alpha</math> such that, whenever <math>x\in{\rm cl}_X(Y)</math> for some subset <math>Y</math> of <math>X,</math> there exists a subset <math>Z</math> of <math>Y</math> with <math>|Z| \leq \alpha,</math> such that <math>x\in \operatorname{cl}_X(Z).</math> Symbolically, <math display=block>t(x, X) = \sup \left\{ \min \{|Z| : Z\subseteq Y\ \wedge\ x\in {\rm cl}_X(Z)\} : Y \subseteq X\ \wedge\ x \in {\rm cl}_X(Y)\right\}.</math> The '''tightness of a space''' <math>X</math> is <math display=block>t(X) = \sup\{t(x, X) : x \in X\}.</math> When <math>t(X) = \aleph_0</math> the space <math>X</math> is said to be ''countably generated'' or ''countably tight''. ** The '''augmented tightness''' of a space <math>X,</math> <math>t^+(X)</math> is the smallest regular cardinal <math>\alpha</math> such that for any <math>Y \subseteq X,</math> <math>x\in{\rm cl}_X(Y)</math> there is a subset <math>Z</math> of <math>Y</math> with cardinality less than <math>\alpha,</math> such that <math>x\in{\rm cl}_X(Z).</math>

===Basic inequalities===

<math display=block>c(X) \leq d(X) \leq w(X) \leq o(X) \leq 2^{|X|}</math> <math display=block>e(X) \leq s(X)</math> <math display=block>\chi(X) \leq w(X)</math> <math display=block>\operatorname{nw}(X) \leq w(X) \text{ and } o(X) \leq 2^{\operatorname{nw}(X)}</math>

==Cardinal functions in Boolean algebras==

Cardinal functions are often used in the study of Boolean algebras.<ref>Monk, J. Donald: ''Cardinal functions on Boolean algebras''. "Lectures in Mathematics ETH Zürich". Birkhäuser Verlag, Basel, 1990. {{isbn|3-7643-2495-3}}.</ref><ref>Monk, J. Donald: ''Cardinal invariants on Boolean algebras''. "Progress in Mathematics", 142. Birkhäuser Verlag, Basel, {{isbn|3-7643-5402-X}}.</ref> We can mention, for example, the following functions:

*'''Cellularity''' <math>c(\mathbb{B})</math> of a Boolean algebra <math>\mathbb{B}</math> is the supremum of the cardinalities of antichains in <math>\mathbb{B}</math>. *'''Length''' <math>{\rm length}(\mathbb{B})</math> of a Boolean algebra <math>\mathbb{B}</math> is ::<math>{\rm length}(\mathbb{B}) = \sup\big\{|A| : A \subseteq \mathbb{B} \text{ is a chain} \big\}</math> *'''Depth''' <math>{\rm depth}(\mathbb{B})</math> of a Boolean algebra <math>\mathbb{B}</math> is ::<math>{\rm depth}(\mathbb{B}) = \sup\big\{|A| : A \subseteq \mathbb{B} \text{ is a well-ordered subset} \big\}</math>. *'''Incomparability''' <math>{\rm Inc}(\mathbb{B})</math> of a Boolean algebra <math>\mathbb{B}</math> is ::<math>{\rm Inc}({\mathbb B}) = \sup\big\{|A| : A \subseteq \mathbb{B} \text{ such that } \forall a,b \in A \big(a \neq b\ \Rightarrow \neg (a\leq b\ \vee \ b \leq a)\big)\big\}</math>. *'''Pseudo-weight''' <math>\pi(\mathbb{B})</math> of a Boolean algebra <math>\mathbb{B}</math> is ::<math>\pi(\mathbb{B}) = \min\big\{|A| : A \subseteq \mathbb{B}\setminus \{0\} \text{ such that } \forall b \in B\setminus\{0\} \big(\exists a \in A\big)\big(a \leq b\big)\big\}.</math>

==Cardinal functions in algebra==

Examples of cardinal functions in algebra are: *Index of a subgroup ''H'' of ''G'' is the number of cosets. *Dimension of a vector space ''V'' over a field ''K'' is the cardinality of any Hamel basis of ''V''. *More generally, for a free module ''M'' over a ring ''R'' we define rank <math>{\rm rank}(M)</math> as the cardinality of any basis of this module. *For a linear subspace ''W'' of a vector space ''V'' we define codimension of ''W'' (with respect to ''V''). *For any algebraic structure it is possible to consider the minimal cardinality of generators of the structure. *For algebraic field extensions, algebraic degree and separable degree 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 is likewise used.

==External links== * A Glossary of Definitions from General Topology [http://math.berkeley.edu/~apollo/topodefs.ps] [https://web.archive.org/web/20120204044336/https://math.berkeley.edu/~apollo/topodefs.ps]

==See also== * Cichoń's diagram

== References == {{reflist}} * {{cite book | last1=Jech | first1=Thomas | author1-link=Thomas Jech | title=Set Theory | edition=Third Millennium | publisher=Springer-Verlag | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-540-44085-7 | year=2003 | zbl=1007.03002 }}

{{DEFAULTSORT:Cardinal Function}} Function Category:Types of functions