# 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 $\mathrm{cf}(\kappa )$ 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
$$.
Replacing $\kappa $ by ${2}^{\kappa}$ in this inequality
we can further deduce that $$.
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^{}).

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 |