cofinality
Definitions
Let be a poset. A subset is said to be cofinal in if for every there is a such that .
A function is said to be cofinal if is cofinal in .
The least cardinality of a cofinal set of is called the cofinality of .
Equivalently, the cofinality of is the least http://planetmath.org/node/2787ordinal
such that there is a cofinal function .
The cofinality of is written , or .
Cofinality of totally ordered sets
If is a totally ordered set, then it must contain a well-ordered cofinal subset which is order-isomorphic to .
Or, put another way, there is a cofinal function with the property that whenever .
For any ordinal we must have , because the identity map on is cofinal. In particular, this is true for cardinals, so any cardinal either satisfies , 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
and are regular cardinals. All other finite cardinals have cofinality and are therefore singular.
It is easy to see that , so is regular.
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 .
In fact, the function given by is cofinal, so .
More generally, for any nonzero limit ordinal , the function given by is cofinal, and this can be used to show that .
Let be an infinite cardinal.
It can be shown that is
the least cardinal such that is
the sum of cardinals each of which is less than .
This fact together with König’s theorem tells us that
.
Replacing by in this inequality
we can further deduce that .
In particular, , from which it follows that (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 |