From: dlrenfro@gateway.net (Dave L. Renfro) Subject: Re: Question from Rudin Date: 11 May 2000 10:36:18 -0400 Newsgroups: sci.math Summary: [missing] paul mitchell [sci.math Wed, 10 May 2000 12:33:09 GMT] wrote > This question from Rudin's textbook, p44 > has been stumping the undergrad Real Analysis > class at UCSC: Is there a non-empty perfect > set in R that contains no rational? We've tried > various translations of the Cantor set, but haven't > been able to demonstrate both closedness and the > absence of rationals. Any hints? Any translation of the Cantor set is closed. There are many ways to prove this. One way is to observe that the Cantor set is compact, any translation is a continuous map, any continuous image of a compact set is compact, and any compact set is closed. [Or: The Cantor set is closed, any translation is a homeomorphism, and any homeomorphic image of a closed set is closed.] In 1884, Scheeffer [4] (pp. 291-293) published a proof of the following: Let C be a perfect nowhere dense set of reals and Z be a countable set of reals. Then for each pair of real numbers a1 < a2, there exists a real number b such that a1 < b < a2 and the b-translate of C has empty intersection with Z. For a proof see pp. 52-53 of Young/Young [5] (these page numbers are for the Chelsea edition). Young/Young's proof actually shows that the set of real numbers b for which the b-translate of C contains no points of Z is the complement of a first category set of real numbers. In particular, there are continuum many such b's in every open interval. Boes/Darst/Erdos prove in [2] that given any countable set Z, the set of b's in (0,1] such that the symmetric Cantor set, formed in the following manner, is disjoint from Z is the complement of a first category subset of (0,1]: Remove from [0,1] a segment of length b/3 so as to leave two intervals of equal length; from each of these two intervals remove a segment of length b/9 to leave 4 intervals of equal length; continue in this manner. In 1959, Bagemihl proved the following stronger version of Scheeffer's theorem. [See also Morgan [3], pp. 194-195.] Let F be a first category set of reals and Z be a countable set of reals. Then the set of real numbers b for which the b-translate of F contains no points of Z is the complement of a first category set of real numbers. [1] Frederick Bagemihl, "A note on Scheeffer's theorem", Michigan Math. J. 2 (1954), 149-150. [2] Duane Boes, Richard Darst, and Paul Erdos, "Fat, Symmetric, Irrational Cantor Sets", The American Mathematical Monthly 88 (1981), 340-341. [3] John C. Morgan II, POINT SET THEORY, Pure and Applied Mathematics 131, Marcel Dekker, 1989. [QA 603 .P67] [4] L. Scheeffer, "Zur Theorie der stetigen Functionen einer reellen Ver�nderlichen", Acta Math. 5 (1884), 279-296. [JFM 16.0340.01 at ] [5] Grace C. Young and William H. Young, THE THEORY OF SETS OF POINTS, Cambridge, 1906. [2'nd edition (actually, just a revised reprint with an appendix of additional notes) was published by Chelsea in 1972, and can be found in most college libraries in the QA 248's.] Dave L. Renfro