Abstract
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 setCλ, for1/3<λ<1, has void intersection with its set of midpoints obtained from distinct members ofCλ.
Highlights
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
We will characterize all points in the unit interval which are not midway between two distinct Cantor points
We shall present a class of Cantor-type sets with the. property that each member of the class has void intersection with its set of distinct midpoints
Summary
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. C. Bose Majumdar [i] have shown that every point in the unit interval is the mean value of a pair of (not necessarily distinct) Cantor ternary points. Jarnik [5] noted that the set of all Cantor Points which represent an irrational number has void intersection with its set of distinct midpoints.
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