Cardinal function

Cardinal function

In mathematics, a cardinal function (or cardinal invariant) is a function that returns cardinal numbers.

Contents

Cardinal functions in set theory

  • The most frequently used cardinal function is a function which assigns to a set its cardinality.
  • 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...".

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.

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