Cardinal arithmetic operations are examples of functions from cardinal numbers (or pairs of them) to cardinal numbers.
Cardinal characteristics of a (proper)
idealI 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 ).
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 ).
Cardinal functions are widely used in
topology as a tool for describing various
topological properties.[2][3] Below are some examples. (Note: some authors, arguing that "there are no finite cardinal numbers in
general topology",[4] 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, for example by adding "" to the right-hand side of the definitions, etc.)
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 (A -base is a set of non-
emptyopen sets whose supersets includes all opens.)
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 hereditary cellularity (sometimes called spread) is the least upper bound of cellularities of its subsets: or where "discrete" means that it is a
discrete topological space.
The extent of a space is So has countable extent exactly when it has no
uncountableclosed 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
Basic inequalities
Cardinal functions in Boolean algebras
Cardinal functions are often used in the study of
Boolean algebras.[5][6] We can mention, for example, the following functions:
Cellularity of a Boolean algebra is the supremum of the cardinalities of
antichains in .
Cardinal arithmetic operations are examples of functions from cardinal numbers (or pairs of them) to cardinal numbers.
Cardinal characteristics of a (proper)
idealI 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 ).
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 ).
Cardinal functions are widely used in
topology as a tool for describing various
topological properties.[2][3] Below are some examples. (Note: some authors, arguing that "there are no finite cardinal numbers in
general topology",[4] 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, for example by adding "" to the right-hand side of the definitions, etc.)
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 (A -base is a set of non-
emptyopen sets whose supersets includes all opens.)
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 hereditary cellularity (sometimes called spread) is the least upper bound of cellularities of its subsets: or where "discrete" means that it is a
discrete topological space.
The extent of a space is So has countable extent exactly when it has no
uncountableclosed 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
Basic inequalities
Cardinal functions in Boolean algebras
Cardinal functions are often used in the study of
Boolean algebras.[5][6] We can mention, for example, the following functions:
Cellularity of a Boolean algebra is the supremum of the cardinalities of
antichains in .