MAT2000 Project: The Cantor Set and its Properties

Ulrik Enstad January 24, 2018

The Cantor set is a special subset of the interval r0, 1s. It has many mysterious properties that challenge our intuition, and is a source for counter-examples in topology and analysis. In terms of size it is uncountable, leading us to consider it “big”. Yet it has Lebesgue measure zero, which means it must be “small” in a sense as well. The Cantor set is an example of a fractal, and it can be used to construct the Cantor function, a function that is continuous with derivative zero almost everywhere, yet takes every value between 0 and 1. Albeit its mysterious nature, the Cantor set appears naturally many places in mathematics, e.g. in dynamics and operator algebras. The goal of this project is to give an exposition of this mysterious set along with its basic properties, and give examples of how it pops up in mathematics. Possibilities include: 1. Construct a function that is continuous with derivative zero almost everywhere, yet takes every value between 0 and 1.

2. Show that if a non-empty metric space is compact, perfect and totally disconnected, then it is topologically the same as the Cantor set. 3. Learn about Hausdorff dimension, and calculate the dimension of the Cantor set. 4. Understand (parts of the) proof that if X is any compact metric space, then there exists a continuous surjection from the Cantor set to X.

Figure 1: Constructing the Cantor set (source: https://en.wikipedia.org/wiki/Cantor_set).

1 Contact information If you think any of these projects sound interesting or have any questions, feel free to drop by my desk at Ullevål or send me an email: • Mail: [email protected]

• My webpage: http://www.mn.uio.no/math/english/people/aca/ubenstad/.

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