A Topological Approach to Selection Rule Stochasticity
Open Access
Author:
Piazza, Jack
Area of Honors:
Mathematics
Degree:
Bachelor of Science
Document Type:
Thesis
Thesis Supervisors:
Jan S Reimann, Thesis Supervisor Sergei Tabachnikov, Thesis Honors Advisor
Keywords:
logic computability topology randomness
Abstract:
Many classical notions of algorithmic randomness are defined using computable selection rules, partial functions from the set of infinite binary strings to itself. While several results are known about these randomness notions, there is no general framework allowing one to study them in a unified way. In this thesis, we introduce a method for using a class of selection rules on a set X to turn X into an Alexandroff topological space. We will first explore point-set topological properties of these spaces, such as connectedness and compactness. We will then investigate homotopy in these spaces and prove a sufficient condition for two of them to be homotopy equivalent. Finally, we will discuss potential applications to classical definitions of stochasticity for infinite binary sequences.
