Cardinal function

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 to cardinal numbers.
• Cardinal characteristics of a ideal I of subsets of X are:
• For a preordered set the bounding number and dominating number are defined as
• In PCF theory the cardinal function is used.

Cardinal functions in topology

• Perhaps the simplest cardinal invariants of a topological space are its cardinality and the cardinality of its topology, denoted respectively by and
• The weight of a topological space is the cardinality of the smallest base for When the space is said to be second countable.
• * The -weight of a space is the cardinality of the smallest -base for
• * The network weight of is the smallest cardinality of a network for A network is a family of sets, for which, for all points and open neighbourhoods containing there exists in for which
• The character of a topological space at a point is the cardinality of the smallest local base for The character of space is When the space is said to be first countable.
• The density of a space is the cardinality of the smallest dense subset of When the space is said to be separable.
• The Lindelöf number of a space is the smallest infinite cardinality such that every open cover has a subcover of cardinality no more than When the space is said to be a Lindelöf space.
• The cellularity or Suslin number of a space is
• The extent of a space is So has countable extent exactly when it has no uncountable closed discrete subset.
• The tightness of a topological space at a point is the smallest cardinal number such that, whenever for some subset of there exists a subset of with such that Symbolically, The tightness of a space is When the space is said to be countably generated or countably tight.
• * The augmented tightness of a space is the smallest regular cardinal such that for any there is a subset of with cardinality less than such that

Cardinal functions in Boolean algebras

• Cellularity of a Boolean algebra is the supremum of the cardinalities of antichains in.
• Length of a Boolean algebra is
• Depth of a Boolean algebra is
• Incomparability of a Boolean algebra is
• Pseudo-weight of a Boolean algebra is

Cardinal functions in algebra

• 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.
• For any algebraic structure it is possible to consider the minimal cardinality of generators of the structure.
• For algebraic field extensions, field extension|algebraic degree] and separable degree are often employed.
• For non-algebraic field extensions, transcendence degree is likewise used.