Cardinal function

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 to cardinal numbers.
• Cardinal characteristics of a ideal I of subsets of X are:
• 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

• 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.
• The weight w of a topological space X is the cardinality of the smallest base for X. When w = 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 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 of a space X is the smallest infinite cardinality such that every open cover has a subcover of cardinality no more than L. When the space X is said to be a Lindelöf space.
• The cellularity or Suslin number of a space X is
• * The hereditary cellularity is the least upper bound of cellularities of its subsets: or with the subspace topology is discrete.
• The tightness t 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 = 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.

  Basic inequalities

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 extensions algebraic degree and separable degree are often employed.
• For non-algebraic field extensions transcendence degree is likewise used.