Hausdorff's theorem

The set of Hausdorff is one of the many mathematical theorems that the German mathematician Felix Hausdorff (1868-1942) to the areas set theory and order theory has contributed. The sentence goes back to Hausdorff's work on confinality and types of order .

Formulation of the sentence

The sentence can be formulated as follows:

In a non-empty linearly ordered set there always exists a subset that is well-ordered by the given order relation and that is confinal in . ${\displaystyle (K,\preccurlyeq )}$ ${\displaystyle \preccurlyeq }$ ${\displaystyle W\subseteq K}$ ${\displaystyle (K,\preccurlyeq )}$

Has the power and has the order type , then the inequality applies to the corresponding initial number . ${\displaystyle K}$ ${\displaystyle |K|=\aleph _{\tau }\;(\tau \in \mathrm {On} )}$ ${\displaystyle W}$ ${\displaystyle \operatorname {ord} (W)=\xi }$ ${\displaystyle \aleph _{\tau }}$ ${\displaystyle \xi \leq \omega _{\tau }}$

Inferences

The following result immediately follows from Hausdorff's theorem:

In a non-empty, partially ordered set, there is always a subset that is well-ordered by the given order relation and with which is confinal in the Hausdorff sense . ${\displaystyle (S,\preccurlyeq )}$ ${\displaystyle \preccurlyeq }$ ${\displaystyle W\subseteq S}$ ${\displaystyle (S,\preccurlyeq )}$

Furthermore, one obtains a result from the theorem using regular ordinal numbers :

Every infinite regular ordinal is an initial number , while the only finite regular ordinals are and . ${\displaystyle \omega _{\tau }}$ ${\displaystyle 0}$ ${\displaystyle 1}$

The sentence also has a further tightening, which essentially goes back to Hausdorff:

For a linearly ordered set , the confinality is always either or or - namely, if it has no largest element - a regular initial number and there is no other regular ordinal number that occurs as an order type of a confinal subset contained in . ${\displaystyle (K,\preccurlyeq )}$ ${\displaystyle \operatorname {cf} (K,\preccurlyeq )}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle (K,\preccurlyeq )}$ ${\displaystyle \operatorname {ord} (W)}$ ${\displaystyle (K,\preccurlyeq )}$ ${\displaystyle W\subseteq K}$

Remarks

• In the proof of Hausdorff's theorem, the well-ordered theorem is essential .
• The proof of the first consequence is based on a direct application of Hausdorff's maximal chain theorem .

literature

• PS Alexandroff : Textbook of set theory . Translated from Russian by Manfred Peschel, Wolfgang Richter and Horst Antelmann. Publishing house Harri Deutsch , Thun and Frankfurt am Main 1994, ISBN 3-8171-1365-X .
• Egbert Harzheim : Ordered Sets (=  Advances in Mathematics . Volume 7 ). Springer Verlag , New York 2005, ISBN 0-387-24219-8 ( MR2127991 ).
• Felix Hausdorff: Investigations into order types I, II, III . In: Reports on the negotiations of the Royal Saxon Society of Sciences . tape 58 , 1906, pp. 106-169 .
• Felix Hausdorff: Investigations on order types IV, V. In: Reports on the negotiations of the Royal Saxon Society of Sciences . tape 59 , 1907, pp. 84-159 .
• Felix Hausdorff: Fundamentals of a theory of ordered sets . In: Mathematical Annals . tape 65 , 1908, pp. 435-505 ( MR1511478 ).
• Felix Hausdorff: Fundamentals of set theory . Reprinted, New York, 1965. Chelsea Publishing Company , New York, NY 1965.
• Erich Kamke : Set theory (=  Göschen Collection . 999 / 999a). 7th edition. Walter de Gruyter , Berlin, New York 1971.
• Wacław Sierpiński : Cardinal and Ordinal Numbers . Panstwowe Wydawnictwo Naukowe, Warsaw 1958 ( MR0095787 ).

Individual evidence

1. ^ PS Alexandroff: Textbook of set theory. 1994, p. 86 ff.
2. ^ Egbert Harzheim: Ordered Sets. 2005, p. 271 ff.
3. ↑ Alexandroff, op.cit., P. 87
4. ↑ ^{a b} Harzheim, op.cit., P. 72.
5. ↑ ^{a b c} Erich Kamke: Set theory. 1971, pp. 167-168.
6. ↑ Harzheim, op.cit., P. 73.
7. ↑ Harzheim, op.cit., P. 74.
8. ^ Wacław Sierpiński: Cardinal and Ordinal Numbers. 1958, pp. 458-459.
9. ↑ Alexandroff, op. Cit., Pp. 88-89