Reducibility and nonreducibility between ℓ^{} equivalence relations
Reducibility and nonreducibility between ℓ^{} equivalence relations
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DOI：
10.1090/s0002-9947-99-02261-8
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发表时间：
1999-01
影响因子：
1.3
通讯作者：
R. Dougherty;G. Hjorth
中科院分区：
文献类型：
--
作者：
R. Dougherty;G. Hjorth
我们证明了，对于1≤p<；q<；∞，无穷实数列之间的`p-等价关系是Borel可约化为`q-等价关系(即商RN/`p的Borel基数不大于RN/`q的Borel基数)，但反之亦然。Borel约化是使用三元Koch雪花曲线的变体构造的；另一个方向的不可约性是通过采用假定的Borel约化，将其提炼成不仅是连续的而且是“模”的约化映射，并使用这个更好的映射来导出矛盾来证明的。
We show that, for 1 ≤ p < q < ∞, the relation of `p-equivalence between infinite sequences of real numbers is Borel reducible to the relation of `q-equivalence (i.e., the Borel cardinality of the quotient RN/`p is no larger than that of RN/`q), but not vice versa. The Borel reduction is constructed using variants of the triadic Koch snowflake curve; the nonreducibility in the other direction is proved by taking a putative Borel reduction, refining it to a reduction map that is not only continuous but ‘modular,’ and using this nicer map to derive a contradiction.