Abstract
This elementary chapter applies the nested intervals theorem to obtain base expansion of real numbers via trees of uniformly subdivided nested closed intervals, with detailed illustrations for ternary expansions. The construction of the Cantor set is then generalized to Cantor systems (systems of nested intervals indexed by binary trees), to formally introduce generalized Cantor sets.
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Our notation is ambiguous since I[1] could denote the second sub interval in two, or three, or ten (or any other numbers) equal subdivisions of I. It would be more correct, but more clumsy, to write \(I_{b}[0],I_{b}[1],\ldots,I_{b}[b - 1]\) in place of \(I[0],I[1],\ldots,I[b - 1]\). Since the base b is generally fixed throughout a situation, it is understood from context, and dropping the subscript b does not cause any confusion.
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Dasgupta, A. (2014). Interval Trees and Generalized Cantor Sets. In: Set Theory. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-8854-5_13
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DOI: https://doi.org/10.1007/978-1-4614-8854-5_13
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Publisher Name: Birkhäuser, New York, NY
Print ISBN: 978-1-4614-8853-8
Online ISBN: 978-1-4614-8854-5
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