ENTIRE FUNCTIONS MAPPING UNCOUNTABLE DENSE SETS OF REALS ONTO EACH OTHER MONOTONICALLY
ENTIRE FUNCTIONS MAPPING UNCOUNTABLE DENSE SETS OF REALS ONTO EACH OTHER MONOTONICALLY
复制标题
将不可数密集实数集单调映射到彼此的整个函数
DOI：
10.1090/s0002-9947-09-04924-1
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发表时间：
2009
影响因子：
1.3
通讯作者：
M. Burke
中科院分区：
文献类型：
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作者：
M. Burke
当A和B是R的可数稠密子集时，Cantor的一个著名结果是A和B是序同构的。K.F.Barth和W.J.Schneider的一个定理指出，序同构可以被认为是非常光滑的，实际上是对整个函数的R的限制。J.E.Baumgartner证明了一致2N0；N1和R的任意两个子集在每个区间内有N1个点是序同构的。然而，U·亚伯拉罕、M·鲁宾和S·谢拉给出了两个这样的集合的ZFC例子，它们的顺序同构不能被认为是光滑的。S.Shelah建立了Baumgartner关于第二类集的结果的一个有用的变体。证明了2N0&>N1和基数N1的第二范畴集的存在是一致的，而与每个区间有第二范畴交的任何两个基数N1集是序同构的.本文证明了谢拉定理中的序同构可以看作是对整函数R的限制。此外，利用L.Hoischen的逼近定理，我们证明了给定一个非负整数n，一个非减的Cn类的g：r→R和一个正的连续函数e：r→R，我们可以选择序同构f，使得对于所有的i=0，1，…，n和对所有的x∈R，|DIf(X)-DIg(X)|<；∈(X).
When A and B are countable dense subsets of R, it is a well-known result of Cantor that A and B are order-isomorphic. A theorem of K.F. Barth and W.J. Schneider states that the order-isomorphism can be taken to be very smooth, in fact the restriction to R of an entire function. J.E. Baumgartner showed that consistently 2 N 0 > N 1 and any two subsets of R having N 1 points in every interval are order-isomorphic. However, U. Abraham, M. Rubin and S. Shelah produced a ZFC example of two such sets for which the order-isomorphism cannot be taken to be smooth. A useful variant of Baumgartner's result for second category sets was established by S. Shelah. He showed that it is consistent that 2 N 0 > N 1 and second category sets of cardinality N 1 exist while any two sets of cardinality N 1 which have second category intersection with every interval are order-isomorphic. In this paper, we show that the order-isomorphism in Shelah's theorem can be taken to be the restriction to R of an entire function. Moreover, using an approximation theorem of L. Hoischen, we show that given a nonnegative integer n, a nondecreasing surjection g: R → R of class C n and a positive continuous function e: R → R, we may choose the order-isomorphism f so that for all i = 0, 1,..., n and for all x ∈ R, |D i f(x)-D i g(x)|<∈(x).