In mathematics, especially in order theory, the cofinality cf(A) of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A. This definition of cofinality relies on the axiom of choice, as it uses the fact that every non-empty set of cardinal numbers has a least member. The cofinality of a partially ordered set A can alternatively be defined as the least ordinal x such that there is a function from x to A with cofinal image.

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  • In mathematics, especially in order theory, the cofinality cf(A) of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A. This definition of cofinality relies on the axiom of choice, as it uses the fact that every non-empty set of cardinal numbers has a least member. The cofinality of a partially ordered set A can alternatively be defined as the least ordinal x such that there is a function from x to A with cofinal image. This second definition makes sense without the axiom of choice. If the axiom of choice is assumed, as will be the case in the rest of this article, then the two definitions are equivalent. Cofinality can be similarly defined for a directed set and is used to generalize the notion of a subsequence in a net.
  • Die Konfinalität bezeichnet in der Mengenlehre eine Eigenschaft von Ordinalzahlen. Der Begriff wurde von Felix Hausdorff eingeführt.
  • Kofinál či také kofinalita limitního ordinálu je matematický pojem z oblasti teorie množin. Je to jedna ze základních charakteristik limitních ordinálů, vyjadřuje „míru přístupnosti horních pater ordinálu“.
  • In teoria degli insiemi, si dice cofinalità di un dato insieme totalmente ordinato <math>I</math> il più piccolo numero ordinale tale che esista una funzione dall'ordinale ad <math>I</math> illimitata. In formule, <math>cof(I) = \min\{\alpha \text{ ordinale } | \exists f:\alpha \rightarrow I\;\; f(\alpha) \text{ è illimitato in }I\} </math> Per illimitato si intende che nessun taglio iniziale di <math>I</math> contiene tutto <math>f(\alpha)</math>, o equivalentemente che dato un qualsiasi elemento <math>x \in I</math> esiste un elemento <math>y \geq x</math> con <math>y \in f(\alpha)</math>. Talvolta si usa, come sinonimo di "illimitato", il termine "cofinale".
  • 極限順序数<math>\alpha</math>の共終数（きょうしゅうすう）とは、 <math>\beta</math>から<math>\alpha</math>への写像で その値域が<math>\alpha</math>の中で非有界になっているようなものが存在するような 最小の<math>\beta</math>のことを言う。 ここで、<math>\alpha</math>の部分集合<math>X</math>が非有界であるとは、 全ての<math>\gamma<\alpha</math>に対して、それよりも大きい <math>X</math>の元が存在することをいう。 また、順序数をそれより小さい順序数全体の集合として定義する公理的集合論の 慣習も採用した。 <math>\alpha</math>の共終数はよく<math>\mbox{cf}(\alpha)</math> と記される。 共終数は順序数の性質として非常に重要なものであり、その他の性質に 大きく影響している。また下記の正則基数と特異基数の違いは顕著である。
  • 在數學裡，尤其是在序理論裡，一个偏序集合 A 的共尾性 cf(A) 是指 A 的共尾子集的勢中的最小者。 共尾性的定義依靠著選擇公理，因为它利用了基數的所有非空集合都有一个最小成员的事实。偏序集合 A 的共尾性亦可定義成最小的序数 x，使得有着值域共尾于陪域的一个从 x 到 A 的函数。第二個定義不需要選擇公理也可以有意義。若假設有選擇公理，接下來的文章將如此假設，這兩種定義將是等價的。 共尾性也可相似地被定義在有向集合上，并且用來廣義化网中的子序列概念。
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  • In mathematics, especially in order theory, the cofinality cf(A) of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A. This definition of cofinality relies on the axiom of choice, as it uses the fact that every non-empty set of cardinal numbers has a least member. The cofinality of a partially ordered set A can alternatively be defined as the least ordinal x such that there is a function from x to A with cofinal image.
  • Die Konfinalität bezeichnet in der Mengenlehre eine Eigenschaft von Ordinalzahlen. Der Begriff wurde von Felix Hausdorff eingeführt.
  • Kofinál či také kofinalita limitního ordinálu je matematický pojem z oblasti teorie množin. Je to jedna ze základních charakteristik limitních ordinálů, vyjadřuje „míru přístupnosti horních pater ordinálu“.
  • In teoria degli insiemi, si dice cofinalità di un dato insieme totalmente ordinato <math>I</math> il più piccolo numero ordinale tale che esista una funzione dall'ordinale ad <math>I</math> illimitata.
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  • Cofinality
  • Konfinalität
  • Kofinál
  • Cofinalità
  • 共終数
  • 共尾性
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