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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 . The existence of Hausdorff gaps shows that the partially ordered set of possible growth rates of sequences is not complete. ==Definition== Let ωω 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 ωω 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 ωω 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 is to replace ωω 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''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hausdorff gap」の詳細全文を読む スポンサード リンク
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