Cardinal function
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In mathematics, a cardinal function is a function that returns cardinal numbers.
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[edit] Cardinal functions in set theory
- The most frequently used cardinal function is a function which assigns to a set its cardinality.
- Cardinal characteristics of an ideal of subsets of X are
- .
- 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 ; if I is a σ-ideal, then add(I)≥.
- .
- 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).
- ,
- The "uniformity number" of I (sometimes also written unif(I)) is the size of the smallest set not in I. By our assumption on I, add(I) ≤ non(I).
-
- 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 the bounding number and dominating number is defined as
-
- ,
- ,
where "" means: "there are infinitely many natural numbers n such that...", and "" means "for all except finitely many natural numbers n we have...".
- In PCF theory the cardinal function [tex]pp_\kappa(\lambda)[/tex] is used.[1]
[edit] Cardinal functions in topology
Cardinal function are widely used in topology as a tool for describing various topological properties[2][3]. For example, the following cardinal functions are used
- Perhaps the simplest cardinal invariant of a topological space X is its cardinality |X|.
- Weight of a space X is is a base for X .
-
- Weight is the minimal cardinality of a basis for X. A topological space X is second countable if and only if w(X)=.
- Density of a space X is .
-
- Density is the minimal cardinality of a dense subset of X. A space X is called separable if d(X)=.
- Cellularity of a space X tis
- Tightness of a space X in a point is
and tightness of a space X is .
-
- The number t(x, X) is the smallest cardinal number α such that, whenever for some subset Y of X, there exists a subset Z having cardinality at most α such that . A space with t(X)= is called countably generated or countably tight.
- Spread of a space X is
[edit] Basic inequalities
- c(X) ≤ d(X) ≤ w(X) ≤ |X|
[edit] Cardinal functions in Boolean algebras
Cardinal functions are often used in the study of Boolean algebras. [4][5]. We can mention, for example, the following functions:
- Cellularity of a Boolean algebra is the supremum of the cardinalities of antichains in .
- Length of a Boolean algebra is
- is a chain
- Depth of a Boolean algebra is
- is a well-ordered subset .
- Incomparability of a Boolean algebra is
- such that .
- Pseudo-weight of a Boolean algebra is
- such that .
[edit] Cardinal functions in algebra
Examples of cardinal functions in algebra are:
- Dimension of a vector space V over a field K is the cardinality of the Hamel basis of V.
- 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.
[edit] See also
[edit] References
- ^ M. Holz, K. Steffens and E. Weitz (1999). Introduction to Cardinal Arithmetic. Birkhäuser. ISBN 3764361247.
- ^ Juhász, István: Cardinal functions in topology. "Mathematical Centre Tracts", nr 34. Mathematisch Centrum, Amsterdam, 1971.
- ^ Juhász, István: Cardinal functions in topology - ten years later. "Mathematical Centre Tracts", 123. Mathematisch Centrum, Amsterdam, 1980. ISBN 90-6196-196-3
- ^ Monk, J. Donald: Cardinal functions on Boolean algebras. "Lectures in Mathematics ETH Zürich". Birkhäuser Verlag, Basel, 1990. ISBN 3-7643-2495-3.
- ^ Monk, J. Donald: Cardinal invariants on Boolean algebras. "Progress in Mathematics", 142. Birkhäuser Verlag, Basel, ISBN 3-7643-5402-X.