Honours Seminar

Location:  Math and Stat Lounge, CW 307.20 

Speaker: Yifan Li

BSc Student Honours in Mathematics supervised by Remus Floricel

Title:  Mappings of the Cantor set and the Hahn–Mazurkiewicz theorem

Abstract:

This presentation focuses on the proof of the Hahn–Mazurkiewicz theorem. In my previous project, I studied examples of space-filling curves, which motivated the question of which spaces can be realized as continuous images of the unit interval. A key step is the result that every compact metric space is a continuous image of the Cantor set. After introducing the construction and basic properties of the Cantor set, we show that all compact, totally disconnected, perfect metric spaces are homeomorphic, and hence homeomorphic to the Cantor set. We then construct a continuous surjection from the Cantor set onto an arbitrary compact metric space. Finally, these results are used to prove the Hahn–Mazurkiewicz theorem, which states that a topological space is a continuous image of the unit interval if and only if it is a Peano space.