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 its cardinality.

• 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 an 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 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).

• For a preordered set $({\mathbb P},\sqsubseteq)$ the bounding number ${\mathfrak b}({\mathbb P})$ and dominating number ${\mathfrak d}({\mathbb P})$ is defined as

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

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

where "$\exists^\infty n\in{\mathbb N}$" means: "there are infinitely many natural numbers n such that...", and "$\forall^\infty n\in{\mathbb N}$" means "for all except finitely many natural numbers n we have...".

• In PCF theory the cardinal function pp_κ(λ) is used.^[1]

Cardinal functions in topology

Cardinal functions are widely used in topology as a tool for describing various topological properties^[2][3]. The cardinal invariants used in topology are usually defined in such a way that excludes finite cardinals numbers. This is sometimes expressed by the phrase "there are no finite cardinal numbers in general topology".^[4]

For example, the following cardinal functions are used

• Perhaps the simplest cardinal invariants of a topological space X are its cardinality |X| and the cardinality of the topology o(X).
• Weight of a space X is ${\rm w}(X)=\min\{|{\mathcal B}|:{\mathcal B}$ is a base for X $\}+\aleph_0$.

Weight is the minimal cardinality of a basis for X. A topological space X is second countable if and only if w(X)=$\aleph_0$.

• • π-weight of a space X is the smallest cardinality of a π-base
• Character of a topological space X at a point x is

$\chi(x,X)=\min\{|{\mathcal B}|:{\mathcal B}$ is a local base for X at x$\}+\aleph_0$ and character of a space X is $\chi(X)=\sup\{\chi(x,X); x\in X\}$,

A topological space is first countable if and only if $\chi(X)=\aleph_0$.

• Density of a space X is ${\rm d}(X)=\min\{|S|:S\subseteq X\ \wedge\ {\rm cl}_X(S)=X \}+\aleph_0$.

Density is the minimal cardinality of a dense subset of X. A space X is called separable if d(X)=$\aleph_0$.

• Cellularity of a space X is

${\rm c}(X)=\sup\{|{\mathcal U}|:{\mathcal U}$ is a family of mutually disjoint non-empty open subsets $X \}+\aleph_0$.

• • Hereditary cellularity (sometimes spread) is the upper bound of cellularities of its subsets:

$s(X)={\rm hc}(X)=\sup\{ {\rm c} Y : Y\subseteq X \}$

or

$s(X)=\sup\{|Y|:Y\subseteq X$ with the subspace topology is discrete }.

• Tightness of a space X in a point $x\in X$ is

$t(x,X)=\sup\big\{\min\{|Z|:Z\subseteq Y\ \wedge\ x\in {\rm cl}_X(Z)\}:Y\subseteq X\ \wedge\ x\in {\rm cl}_X(Y)\big\}$

and tightness of a space X is $t(X)=\sup\{t(x,X):x\in X\}$.

The number t(x, X) is the smallest cardinal number α such that, whenever $x\in{\rm cl}_X(Y)$ for some subset Y of X, there exists a subset Z of Y having cardinality at most α such that $x\in{\rm cl}_X(Z)$. A space with t(X)=$\aleph_0$ is called countably generated or countably tight.

• • Augumented tightness of a space X, t ⁺ (X) is the smallest regular cardinal α such that for any $Y\subseteq X$, $x\in{\rm cl}_X(Y)$ there is a subset Z of Y with cardinality less than α, such that $x\in{\rm cl}_X(Z)$.

Basic inequalities

c(X) ≤ d(X) ≤ w(X) ≤ o(X) ≤ 2^|X|

χ(X) ≤ w(X)

Cardinal functions in Boolean algebras

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

• Cellularity $c({\mathbb B})$ of a Boolean algebra ${\mathbb B}$ is the supremum of the cardinalities of antichains in ${\mathbb B}$.
• Length ${\rm length}({\mathbb B})$ of a Boolean algebra ${\mathbb B}$ is

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

• Depth ${\rm depth}({\mathbb B})$ of a Boolean algebra ${\mathbb B}$ is

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

• Incomparability ${\rm Inc}({\mathbb B})$ of a Boolean algebra ${\mathbb B}$ is

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

• Pseudo-weight $\pi({\mathbb B})$ of a Boolean algebra ${\mathbb B}$ is

$\pi({\mathbb B})=\min\big\{ |A|:A\subseteq {\mathbb B}\setminus \{0\}$ 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:

• 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 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 extensions 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).
• For non-algebraic field extensions transcendence degree is likewise used.

External links

• A Glossary of Definitions from General Topology [1]

See also

Cichoń's diagram

References

1. ^ Holz, Michael; Steffens, Karsten; and Weitz, Edi (1999). Introduction to Cardinal Arithmetic. Birkhäuser. ISBN 3764361247. 
2. ^ Juhász, István: Cardinal functions in topology. "Mathematical Centre Tracts", nr 34. Mathematisch Centrum, Amsterdam, 1971.
3. ^ Juhász, István: Cardinal functions in topology - ten years later. "Mathematical Centre Tracts", 123. Mathematisch Centrum, Amsterdam, 1980. ISBN 90-6196-196-3
4. ^ Engelking, Ryszard (1989). General Topology. Heldermann Verlag, Berlin. ISBN 3885380064. 
5. ^ Monk, J. Donald: Cardinal functions on Boolean algebras. "Lectures in Mathematics ETH Zürich". Birkhäuser Verlag, Basel, 1990. ISBN 3-7643-2495-3.
6. ^ Monk, J. Donald: Cardinal invariants on Boolean algebras. "Progress in Mathematics", 142. Birkhäuser Verlag, Basel, ISBN 3-7643-5402-X.