On Cantor sets

Abstract or Introduction

A good counterexample can be worth just as much as a good theorem. A major counterexample about numerous things in mathematics, that is perfect to have in a back pocket if you want to test out a new definition or statement, is the Cantor set.
It is discovered by Henry John Stephen Smith, firstly introduced by German mathematician Georg Cantor, and also named after him.

The Cantor set is deceptively simple example of a set, that possesses number of fascinating properties, which demonstrate that our intuition about space (even such small space as the unit interval) can be wrong. It is an interesting combination of large and small.

In this thesis, it will be defined several variations of a Cantor set and a few di˙erent constructions of it. We will discuss its properties regardless of the construction and variations, but also some that are not necessarily preserved under constructional transformation.

Another thing that we are going to observe, maybe even more fascinating than set itself, is the Cantor function. It is a great counterexample too, for a lot of theorems and definitions in mathematical analysis. It is a continuous function, that looks like everything, but a continuous function; constant but increasing. The point where things get even more interesting is its derivative.
Also known as Devil’s staircase, Cantor function was also introduced by Georg Cantor ans was later popularised by Schee˙er, Lebesgue and Vitali.