Cardinal function

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

Cardinal functions in set theory[edit]

  • 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.
  • 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\}

Cardinal functions in topology[edit]

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.)

  • Perhaps the simplest cardinal invariants of a topological space X are its cardinality and the cardinality of its topology, denoted respectively by |X | and o(X).
  • The weight w(X ) of a topological space X is the cardinality of the smallest base for X. When w(X ) = \aleph_0 the space X is said to be second countable.
    • The \pi-weight of a space X is the cardinality of the smallest \pi-base for X.
  • The character of a topological space X at a point x is the cardinality of the smallest local base for x. The character of space X is
    \chi(X)=\sup \; \{\chi(x,X) : x\in X\}.
    When \chi(X) = \aleph_0 the space X is said to be first countable.
  • The density d(X ) of a space X is the cardinality of the smallest dense subset of X. When \rm{d}(X) = \aleph_0 the space X is said to be separable.
  • The Lindelöf number L(X ) of a space X is the smallest infinite cardinality such that every open cover has a subcover of cardinality no more than L(X ). When \rm{L}(X) = \aleph_0 the space X is said to be a Lindelöf space.
  • The cellularity of a space X is
    {\rm c}(X)=\sup\{|{\mathcal U}|:{\mathcal U} is a family of mutually disjoint non-empty open subsets of X \}.
    • The Hereditary cellularity (sometimes spread) is the least 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 \}.
  • The tightness t(x, X) of a topological space X at a point x\in X is the smallest cardinal number \alpha such that, whenever x\in{\rm cl}_X(Y) for some subset Y of X, there exists a subset Z of Y, with |Z | ≤ \alpha, such that x\in{\rm cl}_X(Z). Symbolically,
    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\}.
    The tightness of a space X is t(X)=\sup\{t(x,X):x\in X\}. When t(X) = \aleph_0 the space X is said to be countably generated or countably tight.
    • The augmented tightness of a space X, t^+(X) is the smallest regular cardinal \alpha such that for any Y\subseteq X, x\in{\rm cl}_X(Y) there is a subset Z of Y with cardinality less than \alpha, such that x\in{\rm cl}_X(Z).

Basic inequalities[edit]

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

Cardinal functions in Boolean algebras[edit]

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[disambiguation needed] \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[edit]

Examples of cardinal functions in algebra are:

External links[edit]

  • A Glossary of Definitions from General Topology [1]

See also[edit]

Cichoń's diagram

References[edit]

  1. ^ Holz, Michael; Steffens, Karsten; and Weitz, Edi (1999). Introduction to Cardinal Arithmetic. Birkhäuser. ISBN 3764361247. 
  2. ^ Juhász, István (1979). Cardinal functions in topology. Math. Centre Tracts, Amsterdam. ISBN 90-6196-062-2. 
  3. ^ Juhász, István (1980). Cardinal functions in topology - ten years later. 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.