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Cardinal function - Wikipedia

️Tue Mar 18 2014

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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:

${\rm {add}}(I)=\min\{|{\mathcal {A}}|:{\mathcal {A}}\subseteq I\wedge \bigcup {\mathcal {A}}\notin I\}.$

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 $\aleph _{0}$ ; if I is a σ-ideal, then $\operatorname {add} (I)\geq \aleph _{1}.$

$\operatorname {cov} (I)=\min\{|{\mathcal {A}}|:{\mathcal {A}}\subseteq I\wedge \bigcup {\mathcal {A}}=X\}.$

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 ).

$\operatorname {non} (I)=\min\{|A|:A\subseteq X\ \wedge \ A\notin I\},$

The "uniformity number" of I (sometimes also written ${\rm {unif}}(I)$ ) is the size of the smallest set not in I. Assuming I contains all singletons, add(I ) ≤ non(I ).

${\rm {cof}}(I)=\min\{|{\mathcal {B}}|:{\mathcal {B}}\subseteq I\wedge \forall A\in I(\exists B\in {\mathcal {B}})(A\subseteq B)\}.$

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 ).

In the case that $I$ 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.

${\mathfrak {b}}(\mathbb {P} )=\min {\big \{}|Y|:Y\subseteq \mathbb {P} \ \wedge \ (\forall x\in \mathbb {P} )(\exists y\in Y)(y\not \sqsubseteq x){\big \}},$

${\mathfrak {d}}(\mathbb {P} )=\min {\big \{}|Y|:Y\subseteq \mathbb {P} \ \wedge \ (\forall x\in \mathbb {P} )(\exists y\in Y)(x\sqsubseteq y){\big \}}.$

In PCF theory the cardinal function $pp_{\kappa }(\lambda )$ is used.^[1]

Cardinal functions in topology

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Cardinal functions are widely used in topology as a tool for describing various topological properties.^[2]^[3] Below are some examples. (Note: some authors, arguing that "there are no finite cardinal numbers in general topology",^[4] 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 " $\;\;+\;\aleph _{0}$ " to the right-hand side of the definitions, etc.)

$\operatorname {c} (X)=\sup\{|{\mathcal {U}}|:{\mathcal {U}}{\text{ is a family of mutually disjoint non-empty open subsets of }}X\}.$

$c(X)\leq d(X)\leq w(X)\leq o(X)\leq 2^{|X|}$ $e(X)\leq s(X)$ $\chi (X)\leq w(X)$ $\operatorname {nw} (X)\leq w(X){\text{ and }}o(X)\leq 2^{\operatorname {nw} (X)}$

Cardinal functions in Boolean algebras

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Cardinal functions are often used in the study of Boolean algebras.^[5]^[6] We can mention, for example, the following functions:

${\rm {length}}(\mathbb {B} )=\sup {\big \{}|A|:A\subseteq \mathbb {B} {\text{ is a chain}}{\big \}}$

${\rm {depth}}(\mathbb {B} )=\sup {\big \{}|A|:A\subseteq \mathbb {B} {\text{ is a well-ordered subset}}{\big \}}$ .

${\rm {Inc}}({\mathbb {B} })=\sup {\big \{}|A|:A\subseteq \mathbb {B} {\text{ such that }}\forall a,b\in A{\big (}a\neq b\ \Rightarrow \neg (a\leq b\ \vee \ b\leq a){\big )}{\big \}}$ .

$\pi (\mathbb {B} )=\min {\big \{}|A|:A\subseteq \mathbb {B} \setminus \{0\}{\text{ such that }}\forall b\in B\setminus \{0\}{\big (}\exists a\in A{\big )}{\big (}a\leq b{\big )}{\big \}}.$

Cardinal functions in algebra

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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 ${\rm {rank}}(M)$ 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.

A Glossary of Definitions from General Topology [1] [2]

Cichoń's diagram

^ Holz, Michael; Steffens, Karsten; Weitz, Edmund (1999). Introduction to Cardinal Arithmetic. Birkhäuser. ISBN 3764361247.
^ Juhász, István (1979). Cardinal functions in topology (PDF). Math. Centre Tracts, Amsterdam. ISBN 90-6196-062-2. Archived from the original (PDF) on 2014-03-18. Retrieved 2012-06-30.
^ Juhász, István (1980). Cardinal functions in topology - ten years later (PDF). Math. Centre Tracts, Amsterdam. ISBN 90-6196-196-3. Archived from the original (PDF) on 2014-03-17. Retrieved 2012-06-30.
^ Engelking, Ryszard (1989). General Topology. Sigma Series in Pure Mathematics. Vol. 6 (Revised ed.). Heldermann Verlag, Berlin. ISBN 3885380064.
^ Monk, J. Donald: Cardinal functions on Boolean algebras. "Lectures in Mathematics ETH Zürich". Birkhäuser Verlag, Basel, 1990. ISBN 3-7643-2495-3.
^ Monk, J. Donald: Cardinal invariants on Boolean algebras. "Progress in Mathematics", 142. Birkhäuser Verlag, Basel, ISBN 3-7643-5402-X.

Jech, Thomas (2003). Set Theory. Springer Monographs in Mathematics (Third Millennium ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-44085-7. Zbl 1007.03002.