Cofinality

The cofinality (also: cofinality ) is in mathematics a property of ( partial) orders. In set theory, it plays a special role as a property of ordinal and cardinal numbers specifically. The term was introduced by Felix Hausdorff.

Definition

Let be a partially ordered set and through. The set is called cofinal ( kofinal ) in case a with is available for every.

The cofinality of is denoted by and is defined as the smallest cardinality of a confi dimensional subset, ie

For a cardinal number it performs the following terms a: If <, so called singular. If = so called regular.

Properties

  • The cofinality if and only if the partially ordered set is empty.
  • The cofinality if and only if the order has a maximum, such as when there is a Nachfolgerordinalzahl.
  • For non-empty partially ordered sets with no maximum, the cofinality is at least countable, ie (see Aleph function), and at most the cardinality of the set itself, because every partially ordered set is cofinal in itself
  • For totally ordered, ie is regular.
  • For a non - Nachfolgerordinalzahl a subset cofinal if and only if their union equals.
  • Has an infinite amount of regular cardinality, so you need at least - many sets with cardinality less than to represent as the union of these sets.
  • For a limit ordinal a subset cofinal if and only if it converges as a network, provided with the natural order, in the order topology of against.

Examples

  • The cofinality of the natural order is, because the natural numbers form a countable configure dimensional subset.
  • Is regular.
  • Restricting a network under the assumption order to configure dimensional subset a, we obtain a subnet (but not every subnet this shape have ).
  • The cardinal is singular. It is because is configured dimensional subset.
  • Is a Nachfolgerordinalzahl and applies the axiom of choice, it is always regular. The question of whether there is next to another and thus uncountable regular limit cardinal numbers, is the core of the large cardinal axioms, that is, the axioms about the existence of large cardinals.
Aleph number
482004
de