Cofinal (mathematics)

 ( Order Theory ) 

In mathematics, a subset $B\; \backslash subseteq\; A$ of a preordered set $(A,\; \backslash leq)$ is said to be cofinal or frequent in $A$ if for every $a\; \backslash in\; A,$ it is possible to find an element $b$ in $B$ that dominates $a$ (formally, $a\backslash leq\; b$).

Cofinal subsets are very important in the theory of 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 $A$ is referred to as the cofinality of $A.$

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Definitions Let $\backslash ,\backslash leq\backslash ,$ be a homogeneous binary relation on a set $A.$ A subset $B\; \backslash subseteq\; A$ is said to be or with respect to $\backslash ,\backslash leq\backslash ,$ if it satisfies the following condition:

For every $a\; \backslash in\; A,$ there exists some $b\; \backslash in\; B$ that $a\; \backslash leq\; b.$

A subset that is not frequent is called . This definition is most commonly applied when $(A,\; \backslash leq)$ is a directed set, which is a preordered set with additional properties.

Final functions

A map $f\; :\; X\; \backslash to\; A$ between two directed sets is said to be if the image $f(X)$ of $f$ is a cofinal subset of $A.$

Coinitial subsets

A subset $B\; \backslash subseteq\; A$ is said to be (or in the sense of forcing) if it satisfies the following condition:

For every $a\; \backslash in\; A,$ there exists some $b\; \backslash in\; B$ such that $b\; \backslash leq\; a.$

This is the order-theoretic dual to the notion of cofinal subset. Cofinal (respectively coinitial) subsets are precisely the with respect to the right (respectively left) order topology.

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

For a partially ordered set with , every cofinal subset must contain all , otherwise a maximal element that is not in the subset would fail to be 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 form disjoint cofinal subsets of the set of all natural numbers.

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

If $(A,\; \backslash leq)$ is a directed set and if $B\; \backslash subseteq\; A$ is a cofinal subset of $A$ then $(B,\; \backslash leq)$ is also a directed set.

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Examples and sufficient conditions Any superset of a cofinal subset is itself cofinal.

If $(A,\; \backslash leq)$ is a directed set and if some union of (one or more) finitely many subsets $S\_1\; \backslash cup\; \backslash cdots\; \backslash cup\; S\_n$ is cofinal then at least one of the set $S\_1,\; \backslash ldots,\; S\_n$ is cofinal. This property is not true in general without the hypothesis that $(A,\; \backslash leq)$ is directed.

Subset relations and neighborhood bases

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

Cofinal subsets of the real numbers

For any the interval $(x,\; \backslash infty)$ is a cofinal subset of $(\backslash R,\; \backslash leq)$ but it is a cofinal subset of $(\backslash R,\; \backslash geq).$ The set $\backslash N$ of (consisting of positive integers) is a cofinal subset of $(\backslash R,\; \backslash leq)$ but this is true of the set of negative integers $-\; \backslash N\; :=\; \backslash \{\; -1,\; -2,\; -3,\; \backslash ldots\; \backslash \}.$

Similarly, for any the interval $(-\; \backslash infty,\; y)$ is a cofinal subset of $(\backslash R,\; \backslash geq)$ but it is a cofinal subset of $(\backslash R,\; \backslash leq).$ The set $-\; \backslash N$ of negative integers is a cofinal subset of $(\backslash R,\; \backslash geq)$ but this is true of the natural numbers $\backslash N.$ The set $\backslash Z$ of all is a cofinal subset of $(\backslash R,\; \backslash leq)$ and also a cofinal subset of $(\backslash R,\; \backslash geq)$; the same is true of the set $\backslash Q.$

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Cofinal set of subsets A particular but important case is given if $A$ is a subset of the power set $\backslash wp(E)$ of some set $E,$ ordered by reverse inclusion $\backslash ,\backslash supseteq.$ Given this ordering of $A,$ a subset $B\; \backslash subseteq\; A$ is cofinal in $A$ if for every $a\; \backslash in\; A$ there is a $b\; \backslash in\; B$ such that $a\; \backslash supseteq\; b.$

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

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See also

• a subset $U$ of a partially ordered set $(P,\; \backslash leq)$ that contains every element $y\; \backslash in\; P$ for which there is an $x\; \backslash in\; U$ with $x\; \backslash leq\; y$

• edition=3

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