cofinality

Definitions

Let $(P,\leq)$ be a poset. A subset $A\subseteq P$ is said to be cofinal in $P$ if for every $x\in P$ there is a $y\in A$ such that $x\leq y$ . A function $f\colon X\to P$ is said to be cofinal if $f(X)$ is cofinal in $P$ . The least cardinality of a cofinal set of $P$ is called the cofinality of $P$ . Equivalently, the cofinality of $P$ is the least http://planetmath.org/node/2787ordinal $\alpha$ such that there is a cofinal function $f\colon\alpha\to P$ . The cofinality of $P$ is written $\operatorname{cf}(P)$ , or $\operatorname{cof}(P)$ .

Cofinality of totally ordered sets

If $(T,\leq)$ is a totally ordered set, then it must contain a well-ordered cofinal subset which is order-isomorphic to $\operatorname{cf}(T)$ . Or, put another way, there is a cofinal function $f\colon\operatorname{cf}(T)\to T$ with the property that $f(x)<f(y)$ whenever $x<y$ .

For any ordinal $\beta$ we must have $\operatorname{cf}(\beta)\leq\beta$ , because the identity map on $\beta$ is cofinal. In particular, this is true for cardinals, so any cardinal $\kappa$ either satisfies $\operatorname{cf}(\kappa)=\kappa$ , in which case it is said to be regular, or it satisfies $\operatorname{cf}(\kappa)<\kappa$ , in which case it is said to be singular.

The cofinality of any totally ordered set is necessarily a regular cardinal.

Cofinality of cardinals

$0$ and $1$ are regular cardinals. All other finite cardinals have cofinality $1$ and are therefore singular.

It is easy to see that $\operatorname{cf}(\aleph_{0})=\aleph_{0}$ , so $\aleph_{0}$ is regular.

$\aleph_{1}$ is regular, because the union of countably many countable sets is countable. More generally, all infinite successor cardinals are regular.

The smallest infinite singular cardinal is $\aleph_{\omega}$ . In fact, the function $f\colon\omega\to\aleph_{\omega}$ given by $f(n)=\omega_{n}$ is cofinal, so $\operatorname{cf}(\aleph_{\omega})=\aleph_{0}$ . More generally, for any nonzero limit ordinal $\delta$ , the function $f\colon\delta\to\aleph_{\delta}$ given by $f(\alpha)=\omega_{\alpha}$ is cofinal, and this can be used to show that $\operatorname{cf}(\aleph_{\delta})=\operatorname{cf}(\delta)$ .

Let $\kappa$ be an infinite cardinal. It can be shown that $\operatorname{cf}(\kappa)$ is the least cardinal $\mu$ such that $\kappa$ is the sum of $\mu$ cardinals each of which is less than $\kappa$ . This fact together with König’s theorem tells us that $\kappa<\kappa^{\operatorname{cf}(\kappa)}$ . Replacing $\kappa$ by $2^{\kappa}$ in this inequality we can further deduce that $\kappa<\operatorname{cf}(2^{\kappa})$ . In particular, $\operatorname{cf}(2^{\aleph_{0}})>\aleph_{0}$ , from which it follows that $2^{\aleph_{0}}\neq\aleph_{\omega}$ (this being the smallest uncountable aleph which is provably not the cardinality of the continuum).

Title	cofinality
Canonical name	Cofinality
Date of creation	2013-03-22 12:23:55
Last modified on	2013-03-22 12:23:55
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	25
Author	yark (2760)
Entry type	Definition
Classification	msc 03E04
Defines	cofinal
Defines	regular cardinal
Defines	singular cardinal
Defines	regular
Defines	singular