“Hausdorff gap”的意思、由来-开放百科全书

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Hausdorff gap

释义

Definition
References
External links

In mathematics, a Hausdorff gap consists roughly of two collections of sequences of integers, such that there is no sequence lying between the two collections. The first example was found by {{harvs|txt|last=Hausdorff|authorlink=Felix Hausdorff|year=1909}}. The existence of Hausdorff gaps shows that the partially ordered set of possible growth rates of sequences is not complete.

Definition

Let ω^ω be the set of all sequences of non-negative integers, and define f < g to mean lim g(n) – f(n) = +∞.

If X is a poset and κ and λ are cardinals, then a (κ,λ)-pregap in X is a set of elements f_α for α in κ and a set of elements g_β for β in λ such that

The transfinite sequence f is strictly increasing
The transfinite sequence g is strictly decreasing
Every element of the sequence f is less than every element of the sequence g

A pregap is called a gap if it satisfies the additional condition:

There is no element h greater than all elements of f and less than all elements of g.

A Hausdorff gap is a (ω₁,ω₁)-gap in ω^ω such that for every countable ordinal α and every natural number n there are only a finite number of β less than α such that for all k > n we have f_α(k) < g_β(k).

There are some variations of these definitions, with the ordered set ω^ω replaced by a similar set. For example, one can redefine f < g to mean f(n) < g(n) for all but finitely many n. Another variation introduced by {{harvtxt|Hausdorff|1936}} is to replace ω^ω by the set of all subsets of ω, with the order given by A < B if A has only finitely many elements not in B but B has infinitely many elements not in A.

References

{{citation|MR=1311476

|first=Frankiewicz|last= Ryszard|first2= Zbierski|last2= Paweł
|title=Hausdorff gaps and limits
|series=Studies in Logic and the Foundations of Mathematics|volume= 132|publisher= North-Holland Publishing Co.|place= Amsterdam|year= 1994 |isbn= 0-444-89490-X}}

{{citation|title=Die Graduierung nach dem Endverlauf|first=F. |last=Hausdorff

|author-link=Felix Hausdorff
|publisher= B. G. Teubner|year= 1909|series=Abhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig|pages=296−334|volume=31}}

{{citation||title=Summen von ℵ₁ Mengen

|first=F. |last=Hausdorff |publisher=Institute of Mathematics Polish Academy of Sciences
|journal= Fundamenta Mathematicae |issn= 0016-2736
|year=1936
|volume=26
|issue= 1
|pages= 241-255|url=http://matwbn.icm.edu.pl/ksiazki/fm/fm26/fm26126.pdf}}

{{citation|mr=1234288

|last=Scheepers|first= Marion
|chapter=Gaps in ω^ω|title= Set theory of the reals (Ramat Gan, 1991)|pages= 439–561,
|series=Israel Math. Conf. Proc.|volume= 6|publisher= Bar-Ilan Univ.|place= Ramat Gan|year= 1993|editor-first=Haim |editor-last=Judah|isbn=978-9996302800|url=http://www.osti.gov/eprints/topicpages/documents/record/861/4026968.html}}

External links

{{eom|id=Hausdorff_gap}}

2 : Set theory|Order theory

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