On Certain Dense, Uncountable Subsets of the Real Line

Abstract

It is a challenge to define an uncountable set of real numbers that is dense in the real line and whose complement is also uncountable and dense. In this article we specify, for any positive integer k, a partition of the real line into k + 1 sets that are uncountable and dense; moreover, every real number is a condensation point for each of these sets. We show that one of these sets has measure one when restricted to the unit interval, while each of the others has measure zero. We then define a function that is continuous on the set of measure 1 but discontinuous elsewhere.

MSC:

• Primary 26A03
• Secondary 26B05; 28A05

Acknowledgment

We wish to thank the reviewers for their helpful suggestions and the editors for helping us to convert our paper into the Monthly format.

Additional information

Notes on contributors

Chungwu Ho

CHUNGWU HO was born in China, moved to Taiwan in 1949, and came to the U.S. in 1960. He received his training at University of Washington and MIT, where he worked under the guidance of James R. Munkres. He taught for 30 years at Southern Illinois University at Edwardsville, including several years as chair of the math and statistics department. Since retiring there in 2000, he has been teaching at Evergreen Valley College. He writes poetry and plays the Chinese musical instrument, Chinghu, to accompany his wife in Chinese opera singing.

Seth Zimmerman

SETH ZIMMERMAN received his mathematical training at Dartmouth and Princeton. After two decades abroad, teaching in the humanities, he returned to the Bay Area, where by good fortune his office at Evergreen Valley College was adjacent to Chungwu Ho’s, leading to years of fruitful collaboration. He currently pursues his research in mathematics and physics in Bellingham, Washington. His concurrent work includes a popular version of The Inferno of Dante Alighieri and translations of the poems of Mandelstam.