# Cardinal function

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 a function which assigns to a set "A" its cardinality, denoted by | A |.
• 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:
${\rm {add}}(I)=\min\{|{\mathcal {A}}|:{\mathcal {A}}\subseteq I\wedge \bigcup {\mathcal {A}}\notin I{\big \}}$ .
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 add(I)≥$\aleph _{1}$ .
${\rm {cov}}(I)=\min\{|{\mathcal {A}}|:{\mathcal {A}}\subseteq I\wedge \bigcup {\mathcal {A}}=X{\big \}}$ .
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).
${\rm {non}}(I)=\min\{|A|:A\subseteq X\ \wedge \ A\notin I{\big \}}$ ,
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).
${\rm {cof}}(I)=\min\{|{\mathcal {B}}|:{\mathcal {B}}\subseteq I\wedge (\forall A\in I)(\exists B\in {\mathcal {B}})(A\subseteq B){\big \}}.$ 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.
${\mathfrak {b}}({\mathbb {P} })=\min {\big \{}|Y|:Y\subseteq {\mathbb {P} }\ \wedge \ (\forall x\in {\mathbb {P} })(\exists y\in Y)(y\not \sqsubseteq x){\big \}}$ ,
${\mathfrak {d}}({\mathbb {P} })=\min {\big \{}|Y|:Y\subseteq {\mathbb {P} }\ \wedge \ (\forall x\in {\mathbb {P} })(\exists y\in Y)(x\sqsubseteq y){\big \}}$ ## Cardinal functions in topology

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

### Basic inequalities

c(X) ≤ d(X) ≤ w(X) ≤ o(X) ≤ 2|X|
$\chi$ (X) ≤ w(X)

## Cardinal functions in Boolean algebras

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

${\rm {length}}({\mathbb {B} })=\sup {\big \{}|A|:A\subseteq {\mathbb {B} }$ is a chainTemplate:Disambiguation needed ${\big \}}$ ${\rm {depth}}({\mathbb {B} })=\sup {\big \{}|A|:A\subseteq {\mathbb {B} }$ is a well-ordered subset ${\big \}}$ .
${\rm {Inc}}({\mathbb {B} })=\sup {\big \{}|A|:A\subseteq {\mathbb {B} }$ such that ${\big (}\forall a,b\in A{\big )}{\big (}a\neq b\ \Rightarrow \neg (a\leq b\ \vee \ b\leq a){\big )}{\big \}}$ .
$\pi ({\mathbb {B} })=\min {\big \{}|A|:A\subseteq {\mathbb {B} }\setminus \{0\}$ such that ${\big (}\forall b\in B\setminus \{0\}{\big )}{\big (}\exists a\in A{\big )}{\big (}a\leq b{\big )}{\big \}}$ .

## Cardinal functions in algebra

Examples of cardinal functions in algebra are: