cofinality

Definitions

Let $(P,\le )$ 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 $x\le y$.A function $f:X\to P$ 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 $f:\alpha \to P$.The cofinality of $P$ is written $\mathrm{cf}(P)$, or $\mathrm{cof}(P)$.

Cofinality of totally ordered sets

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

For any ordinal $\beta$ we must have $\mathrm{cf}(\beta )\le \beta$, because the identity map on $\beta$ is cofinal.In particular, this is true for cardinals, so any cardinal $\kappa$ either satisfies $\mathrm{cf}(\kappa )=\kappa$, in which case it is said to be regular, or it satisfies , 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 $\mathrm{cf}({\mathrm{\aleph }}_0)={\mathrm{\aleph }}_0$, so ${\mathrm{\aleph }}_0$ is regular.

${\mathrm{\aleph }}_1$ 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 ${\mathrm{\aleph }}_{\omega }$.In fact, the function $f:\omega \to {\mathrm{\aleph }}_{\omega }$ given by $f(n)={\omega }_n$ is cofinal, so $\mathrm{cf}({\mathrm{\aleph }}_{\omega })={\mathrm{\aleph }}_0$.More generally, for any nonzero limit ordinal $\delta$, the function $f:\delta \to {\mathrm{\aleph }}_{\delta }$ given by $f(\alpha )={\omega }_{\alpha }$ is cofinal, and this can be used to show that $\mathrm{cf}({\mathrm{\aleph }}_{\delta })=\mathrm{cf}(\delta )$.

Let $\kappa$ be an infinite cardinal.It can be shown that $\mathrm{cf}(\kappa )$ isthe least cardinal $\mu$ such that $\kappa$ isthe sum of $\mu$ cardinals each of which is less than $\kappa$.This fact together with König’s theorem tells us that.Replacing $\kappa$ by $2^{\kappa }$ in this inequalitywe can further deduce that .In particular, $\mathrm{cf}(2^{{\mathrm{\aleph }}_0})>{\mathrm{\aleph }}_0$, from which it follows that $2^{{\mathrm{\aleph }}_0}\ne {\mathrm{\aleph }}_{\omega }$ (this being the smallest uncountable aleph which is provably not the cardinality of the continuum).