The Cantor Set

Surinder Pal Singh Kainth ()
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Surinder Pal Singh Kainth: Panjab University, Department of Mathematics

Chapter Chapter 10 in A Comprehensive Textbook on Metric Spaces, 2023, pp 279-307 from Springer

Abstract: Abstract This chapter is a detailed treatise on the Cantor set. It starts with a thorough discussion on the basic properties of this set. Then we present a weaker version of Tychonoff’s theorem, which leads to an infinite product representation of the Cantor set. In the next section, we discuss a result of Alexandroff and Hausdorff which states that every complete perfect metric space contains a copy of the Cantor discontinuum. We provide various characterizations of the Cantor space and their applications; including the Brouwer’s theorem which states that every totally disconnected, compact, and perfect metric space is homeomorphic to the Cantor set. We also present a continuous real function that interpolates every bounded sequence of real numbers. This chapter winds up with some miscellaneous topics such as the Cantor function, homeomorphic permutations, and Cantor’s leaky tent.

Date: 2023
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DOI: 10.1007/978-981-99-2738-8_10

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