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In mathematics, the Cantor set, introduced by German mathematician Georg Cantor in 1883^{[1]}^{[2]} (but discovered in 1875 by Henry John Stephen Smith^{[3]}^{[4]}^{[5]}^{[6]}), is a set of points lying on a single line segment that has a number of remarkable and deep properties. Through consideration of it, Cantor and others helped lay the foundations of modern general topology. Although Cantor himself defined the set in a general, abstract way, the most common modern construction is the Cantor ternary set, built by removing the middle thirds of a line segment. Cantor himself only mentioned the ternary construction in passing, as an example of a more general idea, that of a perfect set that is nowhere dense.
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Construction of the ternary set
The Cantor ternary set is created by repeatedly deleting the open middle thirds of a set of line segments. One starts by deleting the open middle third (^{1}⁄_{3}, ^{2}⁄_{3}) from the interval [0, 1], leaving two line segments: [0, ^{1}⁄_{3}] ∪ [^{2}⁄_{3}, 1]. Next, the open middle third of each of these remaining segments is deleted, leaving four line segments: [0, ^{1}⁄_{9}] ∪ [^{2}⁄_{9}, ^{1}⁄_{3}] ∪ [^{2}⁄_{3}, ^{7}⁄_{9}] ∪ [^{8}⁄_{9}, 1]. This process is continued ad infinitum, where the nth set is
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