Entire Functions Mapping Uncountable Dense Sets of Reals onto Each Other Monotonically
نویسنده
چکیده
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 2א0 > א1 and any two subsets of R having א1 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 orderisomorphism 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 2א0 > א1 and second category sets of cardinality א1 exist while any two sets of cardinality א1 which have second category intersection with every interval are order-isomorphic. In this paper, we show that the orderisomorphism 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 Cn and a positive continuous function : R → R, we may choose the order-isomorphism f so that for all i = 0, 1, . . . , n and for all x ∈ R, |Dif(x)−Dig(x)| < (x).
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