Wikipedia / Cardinal function
Wikipedia / Cardinal function
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 the function that 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; if I is a σ-ideal
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) is the size of the smallest set not in I. Assuming I contains all singletons, 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).
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 are defined as
In PCF theory 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"," to the right-hand side of the definitions, etc.)
Perhaps the simplest cardinal invariants of a topological space
The weight is the cardinality of the smallest base is said to be second countable.
The -weight-base is a set of non-empty open sets whose supersets includes all opens.)
The network weight A network is a family and open neighbourhoods
The character at a point is the cardinality of the smallest local base The character is said to be first countable.
The density is the cardinality of the smallest dense subset is said to be separable.
The Lindelöf number is the smallest infinite cardinality such that every open cover is said to be a Lindelöf space.
The cellularity or Suslin number is
The hereditary cellularity (sometimes called spread where "discrete" means that it is a discrete topological space.
The extent has countable extent exactly when it has no uncountable closed discrete subset.
The tightness at a point The tightness of a space is said to be countably generated or countably tight.
The augmented tightness is the smallest regular cardinal
Basic inequalities
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 is the supremum of the cardinalities of antichains.
Length is
Depth is
.
Incomparability is
.
Pseudo-weight is
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 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 generatorss of the structure.
For algebraic field extensions, algebraic degree and separable degree are often employed (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 http://math.berkeley.edu/~apollo/topodefs.ps https://web.archive.org/web/20120204044336/https://math.berkeley.edu/~apollo/topodefs.ps
See also
References