Cardinal function

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

Cardinal functions in set theory

{\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  \operatorname{add}(I) \ge \aleph_1.
\operatorname{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).
\operatorname{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.[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, e.g. by adding "\;\; + \;\aleph_0" to the right-hand side of the definitions, etc.)

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

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.[5][6] We can mention, for example, the following functions:

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

External links

See also

References

  1. Holz, Michael; Steffens, Karsten; and Weitz, Edi (1999). Introduction to Cardinal Arithmetic. Birkhäuser. ISBN 3764361247.
  2. Juhász, István (1979). Cardinal functions in topology (PDF). Math. Centre Tracts, Amsterdam. ISBN 90-6196-062-2.
  3. Juhász, István (1980). Cardinal functions in topology - ten years later (PDF). Math. Centre Tracts, Amsterdam. ISBN 90-6196-196-3.
  4. Engelking, Ryszard (1989). General Topology. Sigma Series in Pure Mathematics 6 (Revised ed.). 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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