View source for Cofinal (mathematics)

{{Short description|Mathematical property of subsets in order theory}}
In [[Mathematics|mathematics]], a [[Subset|subset]] <math>B \subseteq A</math> of a preordered set <math>(A, \leq)</math> is said to be '''cofinal''' or '''frequent'''{{sfn|Schechter|1996|pp=158-165}} in <math>A</math> if for every <math>a \in A,</math> it is possible to find an element <math>b</math> in <math>B</math> that is "larger than <math>a</math>" (explicitly, "larger than <math>a</math>" means <math>a \leq b</math>). 

Cofinal subsets are very important in the theory of [[Directed set|directed set]]s and [[Net (mathematics)|nets]], where “[[Subnet (mathematics)|cofinal subnet]]” is the appropriate generalization of "[[Subsequence|subsequence]]".  They are also important in [[Order theory|order theory]], including the theory of cardinal numbers, where the minimum possible [[Cardinality|cardinality]] of a cofinal subset of <math>A</math> is referred to as the [[Cofinality|cofinality]] of <math>A.</math>

== Definitions ==

Let <math>\,\leq\,</math> be a homogeneous binary relation on a set <math>A.</math> 
A subset <math>B \subseteq A</math> is said to be {{em|cofinal}} or {{em|frequent}}{{sfn|Schechter|1996|pp=158-165}} with respect to <math>\,\leq\,</math> if it satisfies the following condition: 
:For every <math>a \in A,</math> there exists some <math>b \in B</math> that <math>a \leq b.</math> 

A subset that is not frequent is called {{em|infrequent}}.{{sfn|Schechter|1996|pp=158-165}} 
This definition is most commonly applied when <math>(A, \leq)</math> is a [[Directed set|directed set]], which is a preordered set with additional properties. 

;Final functions

A [[Function (mathematics)|map]] <math>f : X \to A</math> between two directed sets is said to be {{em|final}}<ref name=bredon93>{{cite book|last=Bredon|first=Glen|title=Topology and Geometry|year=1993|publisher=Springer|page=16}}</ref> if the image <math>f(X)</math> of <math>f</math> is a cofinal subset of <math>A.</math>

;Coinitial subsets

A subset <math>B \subseteq A</math> is said to be {{em|coinitial}} (or {{em|dense}} in the sense of [[Forcing (mathematics)|forcing]]) if it satisfies the following condition: 

:For every <math>a \in A,</math> there exists some <math>b \in B</math> such that <math>b \leq a.</math> 

This is the [[Duality (order theory)|order-theoretic dual]] to the notion of cofinal subset. 
Cofinal (respectively coinitial) subsets are precisely the [[Dense set|dense set]]s with respect to the right (respectively left) [[Order topology#Left and right order topologies|order topology]].

== Properties ==

The cofinal relation over partially ordered sets ("[[Partially ordered set|poset]]s") is [[Reflexive relation|reflexive]]: every poset is cofinal in itself. It is also [[Transitive relation|transitive]]: if <math>B</math> is a cofinal subset of a poset <math>A,</math> and <math>C</math> is a cofinal subset of <math>B</math> (with the partial ordering of <math>A</math> applied to <math>B</math>), then <math>C</math> is also a cofinal subset of <math>A.</math> 

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 {{em|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 number|natural number]]s form disjoint cofinal subsets of the set of all natural numbers. 

If a partially ordered set <math>A</math> admits a totally ordered cofinal subset, then we can find a subset <math>B</math> that is well-ordered and cofinal in <math>A.</math> 

If <math>(A, \leq)</math> is a [[Directed set|directed set]] and if <math>B \subseteq A</math> is a cofinal subset of <math>A</math> then <math>(B, \leq)</math> is also a directed set.{{sfn|Schechter|1996|pp=158-165}} 

== Examples and sufficient conditions ==

Any superset of a cofinal subset is itself cofinal.{{sfn|Schechter|1996|pp=158-165}} 

If <math>(A, \leq)</math> is a directed set and if some union of (one or more) finitely many subsets <math>S_1 \cup \cdots \cup S_n</math> is cofinal then at least one of the set <math>S_1, \ldots, S_n</math> is cofinal.{{sfn|Schechter|1996|pp=158-165}} This property is not true in general without the hypothesis that <math>(A, \leq)</math> is directed.

;Subset relations and neighborhood bases

Let <math>X</math> be a [[Topological space|topological space]] and let <math>\mathcal{N}_x</math> denote the neighborhood filter at a point <math>x \in X.</math> 
The superset relation <math>\,\supseteq\,</math> is a partial order on <math>\mathcal{N}_x</math>: explicitly, for any sets <math>S</math> and <math>T,</math> declare that <math>S \leq T</math> if and only if <math>S \supseteq T</math> (so in essence, <math>\,\leq\,</math> is equal to <math>\,\supseteq\,</math>). 
A subset <math>\mathcal{B} \subseteq \mathcal{N}_x</math> is called a {{em|neighborhood base}} at <math>x</math> if (and only if) <math>\mathcal{B}</math> is a cofinal subset of <math>\left(\mathcal{N}_x, \supseteq\right);</math> 
that is, if and only if for every <math>N \in \mathcal{N}_x</math> there exists some <math>B \in \mathcal{B}</math> such that <math>N \supseteq B.</math> (I.e. such that <math>N \leq B</math>.)

;Cofinal subsets of the real numbers

For any <math>- \infty < x < \infty,</math> the interval <math>(x, \infty)</math> is a cofinal subset of <math>(\R, \leq)</math> but it is {{em|not}} a cofinal subset of <math>(\R, \geq).</math> 
The set <math>\N</math> of [[Natural number|natural number]]s (consisting of positive integers) is a cofinal subset of <math>(\R, \leq)</math> but this is {{em|not}} true of the set of negative integers <math>- \N := \{ -1, -2, -3, \ldots \}.</math> 

Similarly, for any <math>-\infty < y < \infty,</math> the interval <math>(- \infty, y)</math> is a cofinal subset of <math>(\R, \geq)</math> but it is {{em|not}} a cofinal subset of <math>(\R, \leq).</math> 
The set <math>- \N</math> of negative integers is a cofinal subset of <math>(\R, \geq)</math> but this is {{em|not}} true of the natural numbers <math>\N.</math> 
The set <math>\Z</math> of all [[Integer|integer]]s is a cofinal subset of <math>(\R, \leq)</math> and also a cofinal subset of <math>(\R, \geq)</math>; the same is true of the set <math>\Q.</math>

== Cofinal set of subsets ==

A particular but important case is given if <math>A</math> is a subset of the [[Power set|power set]] <math>\wp(E)</math> of some set <math>E,</math> ordered by reverse inclusion <math>\,\supseteq.</math> Given this ordering of <math>A,</math> a subset <math>B \subseteq A</math> is cofinal in <math>A</math> if for every <math>a \in A</math> there is a <math>b \in B</math> such that <math>a \supseteq b.</math> 

For example, let <math>E</math> be a [[Group (mathematics)|group]] and let <math>A</math> be the set of [[Normal subgroup|normal subgroup]]s of finite [[Index of a subgroup|index]]. The [[Profinite group|profinite completion]] of <math>E</math> is defined to be the [[Inverse limit|inverse limit]] of the [[Inverse system|inverse system]] of finite quotients of <math>E</math> (which are parametrized by the set <math>A</math>). 
In this situation, every cofinal subset of <math>A</math> is sufficient to construct and describe the profinite completion of <math>E.</math> 

== See also ==

* {{annotated link|Cofinality}}
* {{annotated link|Upper set}} 
** a subset <math>U</math> of a partially ordered set <math>(P, \leq)</math> that contains every element <math>y \in P</math> for which there is an <math>x \in U</math> with <math>x \leq y</math>

==References==

{{reflist}}

* {{Lang Algebra|edition=3}}
* {{Schechter Handbook of Analysis and Its Foundations}} <!-- {{sfn|Schechter|1996|p=}} -->


[[Category:Order theory]]

{{Sourceattribution|Cofinal (mathematics)}}