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| In mathematics, a '''cardinal function''' (or '''cardinal invariant''') is a function that returns [[cardinal number]]s.
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| == Cardinal functions in set theory ==
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| * The most frequently used cardinal function is a function which assigns to a [[Set (mathematics)|set]] "A" its [[cardinality]], denoted by | ''A'' |.
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| * [[Aleph number]]s and [[beth number]]s can both be seen as cardinal functions defined on [[ordinal number]]s.
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| * [[Cardinal arithmetic]] operations are examples of functions from cardinal numbers (or pairs of them) to cardinal numbers.
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| * Cardinal characteristics of a (proper) [[ideal (set theory)|ideal]] ''I'' of subsets of ''X'' are:
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| :<math>{\rm add}(I)=\min\{|{\mathcal A}|: {\mathcal A}\subseteq I \wedge \bigcup{\mathcal A}\notin I\big\}</math>.
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| ::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 σ-ideal, then add(''I'')≥<math>\aleph_1</math>.
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| :<math>{\rm cov}(I)=\min\{|{\mathcal A}|:{\mathcal A}\subseteq I \wedge\bigcup{\mathcal A}=X\big\}</math>.
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| :: 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'').
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| :<math>{\rm non}(I)=\min\{|A|:A\subseteq X\ \wedge\ A\notin I\big\}</math>,
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| :: 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'') ≤ non(''I'').
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| :<math>{\rm cof}(I)=\min\{|{\mathcal B}|:{\mathcal B}\subseteq I \wedge (\forall A\in I)(\exists B\in {\mathcal B})(A\subseteq B)\big\}.</math>
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| :: The "cofinality" of ''I'' is the [[cofinality]] of the [[partial order]] (''I'', ⊆). It is easy to see that we must have non(''I'') ≤ cof(''I'') and cov(''I'') ≤ cof(''I'').
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| :In the case that <math>I</math> is an ideal closely related to the structure of the reals, such as the ideal of [[Null set#Lebesgue measure|Lebesgue null sets]] or the ideal of [[meagre set]]s, these cardinal invariants are referred to as [[Cardinal characteristic of the continuum|cardinal characteristics of the continuum]].
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| * 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> is defined as
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| ::<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>,
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| ::<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><!--,
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| 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...". -->
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| <!-- 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.-->
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| * In [[PCF theory]] the cardinal function <math>pp_\kappa(\lambda)</math> is used.<ref>{{cite book | author=Holz, Michael; Steffens, Karsten; and Weitz, Edi | title=Introduction to Cardinal Arithmetic | publisher=Birkhäuser | year=1999 | isbn=3764361247}}</ref>
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| == Cardinal functions in topology ==
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| Cardinal functions are widely used in [[topology]] as a tool for describing various [[topological properties]].<ref>{{cite book | last=Juhász | first=István | 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}}</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}}</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 | note=Revised and completed edition, Sigma Series in Pure Mathematics, Vol. 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, e.g. by adding "<math>\;\; + \;\aleph_0</math>" to the right-hand side of the definitions, etc.)
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| * Perhaps the simplest cardinal invariants of a topological space ''X'' are its cardinality and the cardinality of its topology, denoted respectively by |''X'' | and ''o''(''X'').
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| * The '''[[Base (topology)#Weight and character|weight]]''' w(''X'' ) of a topological space ''X'' is the cardinality of the smallest [[Base (topology)|base]] for ''X''. When w(''X'' ) = <math>\aleph_0</math> the space ''X'' is said to be ''[[second countable]]''.
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| ** The '''<math>\pi</math>-weight''' of a space ''X'' is the cardinality of the smallest <math>\pi</math>-base for ''X''.
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| * The '''[[Base (topology)#Weight and character|character]]''' of a topological space ''X'' '''at a point''' ''x'' is the cardinality of the smallest [[Neighbourhood system|local base]] for ''x''. The '''character''' of space ''X'' is <center><math>\chi(X)=\sup \; \{\chi(x,X) : x\in X\}.</math></center> When <math>\chi(X) = \aleph_0</math> the space ''X'' is said to be ''[[First-countable space|first countable]]''.
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| * The '''density''' d(''X'' ) of a space ''X'' is the cardinality of the smallest dense subset of ''X''. When <math>\rm{d}(X) = \aleph_0</math> the space ''X'' is said to be ''[[Separable space|separable]]''.
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| * The '''[[Lindelöf space#Generalisation|Lindelöf number]]''' L(''X'' ) of a space ''X'' is the smallest infinite cardinality such that every [[open cover]] has a subcover of cardinality no more than L(''X'' ). When <math>\rm{L}(X) = \aleph_0</math> the space ''X'' is said to be a ''[[Lindelöf space]]''.
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| * The '''cellularity''' of a space ''X'' is <center><math>{\rm c}(X)=\sup\{|{\mathcal U}|:{\mathcal U}</math> is a [[family of sets|family]] of mutually [[disjoint sets|disjoint]] non-empty [[open set|open]] subsets of <math>X \}</math>.</center>
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| ** The '''Hereditary cellularity''' (sometimes '''spread''') is the least upper bound of cellularities of its subsets: <center><math>s(X)={\rm hc}(X)=\sup\{ {\rm c} (Y) : Y\subseteq X \}</math></center> or <center><math>s(X)=\sup\{|Y|:Y\subseteq X </math> with the [[subspace (topology)|subspace]] topology is [[discrete topological space|discrete]] <math>\}</math>.</center>
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| * The '''tightness''' ''t''(''x'', ''X'') of a topological space ''X'' '''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 ''Y'' of ''X'', there exists a subset ''Z'' of ''Y'', with |''Z'' | ≤ <math>\alpha</math>, such that <math>x\in{\rm cl}_X(Z)</math>. Symbolically, <center><math>t(x,X)=\sup\big\{\min\{|Z|:Z\subseteq Y\ \wedge\ x\in {\rm cl}_X(Z)\}:Y\subseteq X\ \wedge\ x\in {\rm cl}_X(Y)\big\}.</math></center> The '''tightness of a space''' ''X'' is <math>t(X)=\sup\{t(x,X):x\in X\}</math>. When ''t(X) = ''<math>\aleph_0</math> the space ''X'' is said to be ''[[countably generated space|countably generated]]'' or ''[[countable tightness|countably tight]]''.
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| ** The '''augmented tightness''' of a space ''X'', <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 ''Z'' of ''Y'' with cardinality less than <math>\alpha</math>, such that <math>x\in{\rm cl}_X(Z)</math>.
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| ===Basic inequalities===
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| : ''c''(''X'') ≤ ''d''(''X'') ≤ ''w''(''X'') ≤ ''o''(''X'') ≤ 2<sup>|X|</sup>
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| : <math>\chi</math>(''X'') ≤ ''w''(''X'')
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| ==Cardinal functions in Boolean algebras==
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| Cardinal functions are often used in the study of [[Boolean algebra (logic)|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:
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| *'''Cellularity''' <math>c({\mathbb B})</math> of a Boolean algebra <math>{\mathbb B}</math> is the supremum of the cardinalities of [[antichain]]s in <math>{\mathbb B}</math>.
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| *'''Length''' <math>{\rm length}({\mathbb B})</math> of a Boolean algebra <math>{\mathbb B}</math> is
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| :<math>{\rm length}({\mathbb B})=\sup\big\{|A|:A\subseteq {\mathbb B}</math> is a [[chain (mathematics)|chain]]{{dn|date=May 2013}} <math>\big\}</math>
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| *'''Depth''' <math>{\rm depth}({\mathbb B})</math> of a Boolean algebra <math>{\mathbb B}</math> is
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| :<math>{\rm depth}({\mathbb B})=\sup\big\{ |A|:A\subseteq {\mathbb B}</math> is a [[well-ordered]] subset <math>\big\}</math>.
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| *'''Incomparability''' <math>{\rm Inc}({\mathbb B})</math> of a Boolean algebra <math>{\mathbb B}</math> is
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| :<math>{\rm Inc}({\mathbb B})=\sup\big\{ |A|:A\subseteq {\mathbb B}</math> such that <math>\big(\forall a,b\in A\big)\big(a\neq b\ \Rightarrow \neg (a\leq b\ \vee \ b\leq a)\big)\big\}</math>.
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| *'''Pseudo-weight''' <math>\pi({\mathbb B})</math> of a Boolean algebra <math>{\mathbb B}</math> is
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| :<math>\pi({\mathbb B})=\min\big\{ |A|:A\subseteq {\mathbb B}\setminus \{0\}</math> such that <math>\big(\forall b\in B\setminus \{0\}\big)\big(\exists a\in A\big)\big(a\leq b\big)\big\}</math>.
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| ==Cardinal functions in algebra==
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| Examples of cardinal functions in algebra are:
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| *[[Index of a subgroup]] ''H'' of ''G'' is the number of cosets.
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| *Dimension of a [[vector space]] ''V'' over a [[field (mathematics)|field]] ''K'' is the cardinality of any [[Hamel basis]] of ''V''.
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| *More generally, for a free [[module (mathematics)|module]] ''M'' over a [[ring (mathematics)|ring]] ''R'' we define rank <math>{\rm rank}(M)</math> as the cardinality of any basis of this module.
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| *For a [[linear subspace]] ''W'' of a vector space ''V'' we define [[codimension]] of ''W'' (with respect to ''V'').
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| *For any [[algebraic structure]] it is possible to consider the minimal cardinality of generators of the structure.
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| *For [[algebraic extension]]s [[Degree of a field extension|algebraic degree]] 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).
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| *For non-algebraic [[field extension]]s [[transcendence degree]] is likewise used.
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| ==External links==
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| * A Glossary of Definitions from General Topology [http://math.berkeley.edu/~apollo/topodefs.ps]
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| ==See also==
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| [[Cichoń's diagram]]
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| == References ==
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| <references/>
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| {{DEFAULTSORT:Cardinal Function}}
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| [[Category:Cardinal numbers]]
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