Cofinal (mathematics)

In mathematics, a subset of a preordered set is said to be cofinal or frequent[1] in if for every it is possible to find an element in that is "larger than " (explicitly, "larger than " means ).

Cofinal subsets are very important in the theory of directed sets and nets, where “cofinal subnet” is the appropriate generalization of "subsequence". They are also important in order theory, including the theory of cardinal numbers, where the minimum possible cardinality of a cofinal subset of is referred to as the cofinality of

DefinitionsEdit

Let   be a homogeneous binary relation on a set   A subset   is said to be cofinal or frequent[1] with respect to   if it satisfies the following condition:

For every   there exists some   that  

A subset that is not frequent is called infrequent.[1] This definition is most commonly applied when   is a directed set, which is a preordered set with additional properties.

Final functions

A map   between two directed sets is said to be final[2] if the image   of   is a cofinal subset of  

Coinitial subsets

A subset   is said to be coinitial (or dense in the sense of forcing) if it satisfies the following condition:

For every   there exists some   such that  

This is the order-theoretic dual to the notion of cofinal subset. Cofinal and coinitial subsets are both dense in the sense of the appropriate (right or left) order topology.

PropertiesEdit

The cofinal relation over partially ordered sets ("posets") is reflexive: every poset is cofinal in itself. It is also transitive: if   is a cofinal subset of a poset   and   is a cofinal subset of   (with the partial ordering of   applied to  ), then   is also a cofinal subset of  

For a partially ordered set with maximal elements, every cofinal subset must contain all maximal elements, otherwise a maximal element that is not in the subset would fail to be less than or equal to any element of the subset, violating the definition of cofinal. For a partially ordered set with a greatest element, a subset is cofinal if and only if it contains that greatest element (this follows, since a greatest element is necessarily a maximal element). Partially ordered sets without greatest element or maximal elements admit disjoint cofinal subsets. For example, the even and odd natural numbers form disjoint cofinal subsets of the set of all natural numbers.

If a partially ordered set   admits a totally ordered cofinal subset, then we can find a subset   that is well-ordered and cofinal in  

If   is a directed set and if   is a cofinal subset of   then   is also a directed set.[1]

Examples and sufficient conditionsEdit

Any superset of a cofinal subsets is itself cofinal.[1]

If   is a preordered set and if some union of (one or more) finitely many subsets   is cofinal then at least one of the set   is cofinal.[1]

Subset relations and neighborhood bases

Let   be a topological space and let   denotes the neighborhood filter at a point   The superset relation   is a partial order on  : explicitly, for any sets   and   declare that   if and only if   (so in essence,   is equal to  ). A subset   is called a neighborhood base at   if (and only if)   is a cofinal subset of   that is, if and only if for every   there exists some   such that   (i.e. such that  ).

Cofinal subsets of the real numbers

For any   the interval   is a cofinal subset of   but it is not a cofinal subset of   The set   of natural numbers (consisting of positive integers) is a cofinal subset of   but this is not true of the set of negative integers  

Similarly, for any   the interval   is a cofinal subset of   but it is not a cofinal subset of   The set   of negative integers is a cofinal subset of   but this is not true of the natural numbers   The set   of all integers is a cofinal subset of   and also a cofinal subset of  ; the same is true of the set  

Cofinal set of subsetsEdit

A particular but important case is given if   is a subset of the power set   of some set   ordered by reverse inclusion   Given this ordering of   a subset   is cofinal in   if for every   there is a   such that  

For example, let   be a group and let   be the set of normal subgroups of finite index. The profinite completion of   is defined to be the inverse limit of the inverse system of finite quotients of   (which are parametrized by the set  ). In this situation, every cofinal subset of   is sufficient to construct and describe the profinite completion of  

See alsoEdit

  • Cofinite
  • Cofinality – Size of subsets in order theory
  • Upper set – Subset of a preordered space that contains any element that is larger than any one element of the subset
    • a subset   of a partially ordered set   that contains every element   for which there is an   with  

ReferencesEdit

  1. ^ a b c d e f Schechter 1996, pp. 158-165.
  2. ^ Bredon, Glen (1993). Topology and Geometry. Springer. p. 16.