Let (P,) be a poset. A subset AP is said to be cofinalPlanetmathPlanetmath in P if for every xP there is a yA such that xy. A function f:XP 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 α such that there is a cofinal function f:αP. The cofinality of P is written cf(P), or cof(P).

Cofinality of totally ordered sets

If (T,) is a totally ordered setMathworldPlanetmath, then it must contain a well-ordered cofinal subset which is order-isomorphic to cf(T). Or, put another way, there is a cofinal function f:cf(T)T with the property that f(x)<f(y) whenever x<y.

For any ordinal β we must have cf(β)β, because the identity map on β is cofinal. In particular, this is true for cardinals, so any cardinal κ either satisfies cf(κ)=κ, in which case it is said to be regular, or it satisfies cf(κ)<κ, 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 cf(0)=0, so 0 is regular.

1 is regular, because the union of countably many countable sets is countableMathworldPlanetmath. More generally, all infiniteMathworldPlanetmath successor cardinals are regular.

The smallest infinite singular cardinal is ω. In fact, the function f:ωω given by f(n)=ωn is cofinal, so cf(ω)=0. More generally, for any nonzero limit ordinalMathworldPlanetmath δ, the function f:δδ given by f(α)=ωα is cofinal, and this can be used to show that cf(δ)=cf(δ).

Let κ be an infinite cardinal. It can be shown that cf(κ) is the least cardinal μ such that κ is the sum of μ cardinals each of which is less than κ. This fact together with König’s theorem tells us that κ<κcf(κ). Replacing κ by 2κ in this inequality we can further deduce that κ<cf(2κ). In particular, cf(20)>0, from which it follows that 20ω (this being the smallest uncountable aleph which is provably not the cardinality of the continuumMathworldPlanetmath).

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