International Journal of Mathematics and Mathematical Sciences
Volume 1 (1978), Issue 4, Pages 541-546

The midpoint set of a cantor set

Ken W. Lee

Department of Mathematical Sciences, Missouri Western State College, 4525 Downs Drive, Saint Joseph 64507, Missouri, USA

Received 28 March 1978

Copyright © 1978 Ken W. Lee. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A non-endpoint of the Cantor ternary set is any Cantor point which is not an endpoint of one of the remaining closed intervals obtained in the usual construction process of the Cantor ternary set in the unit interval. It is shown that the set of points in the unit interval which are not midway between two distinct Cantor ternary points is precisely the set of Cantor nonendpoints. It is also shown that the generalized Cantor set Cλ, for 1/3<λ<1, has void intersection with its set of midpoints obtained from distinct members of Cλ.