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 "A" its cardinality, denoted by | A |.
- 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 a (proper) ideal I 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
- 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 ) is the size of the smallest set not in I. Assuming I contains all singletons, add(I) ≤ non(I).
- In the case that 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 the bounding number and dominating number is defined as
- In PCF theory the cardinal function is used.
Cardinal functions in topology
Cardinal functions are widely used in topology as a tool for describing various topological properties. Below are some examples. (Note: some authors, arguing that "there are no finite cardinal numbers in general topology", 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 "" 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 ) = the space X is said to be second countable.
- The -weight of a space X is the cardinality of the smallest -base for X.
- The network weight of X is the smallest cardinality of a network for X. A network is a family of sets, for which, for all points x and open neighbourhoods U containing x, there exists B in for which x ∈ B ⊆ U.
- 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
When 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 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 the space X is said to be a Lindelöf space.
- The cellularity or Suslin number of a space X is
- The tightness t(x, X) of a topological space X at a point is the smallest cardinal number such that, whenever for some subset Y of X, there exists a subset Z of Y, with |Z | ≤ , such that . Symbolically,
The tightness of a space X is . When t(X) = the space X is said to be countably generated or countably tight.
- The augmented tightness of a space X, is the smallest regular cardinal such that for any , there is a subset Z of Y with cardinality less than , such that .
- c(X) ≤ d(X) ≤ w(X) ≤ o(X) ≤ 2|X|
- (X) ≤ w(X)
- nw(X) ≤ w(X) and o(X) ≤ 2nw(X)
Cardinal functions in Boolean algebras
Cardinal functions are often used in the study of Boolean algebras. 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
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 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.
- Holz, Michael; Steffens, Karsten; Weitz, Edmund (1999). Introduction to Cardinal Arithmetic. Birkhäuser. ISBN 3764361247.
- Juhász, István (1979). Cardinal functions in topology (PDF). Math. Centre Tracts, Amsterdam. ISBN 90-6196-062-2.
- Juhász, István (1980). Cardinal functions in topology - ten years later (PDF). Math. Centre Tracts, Amsterdam. ISBN 90-6196-196-3.
- Engelking, Ryszard (1989). General Topology. Sigma Series in Pure Mathematics. 6 (Revised ed.). Heldermann Verlag, Berlin. ISBN 3885380064.
- 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.