(This was the basis for a talk I gave at the Toronto Student Set Theory and Topology seminar on January 15, 2013. The assumed knowledge is an undergraduate course in general topology. This is only a draft, and will be updated soon.)

## Introduction

There are many questions in mathematics and sciences in general whose main object of study is the topological group. These objects are very versatile and can represent many of the structures we encounter. One question that I’ve been working on examines the (extreme) dynamical properties of topologies on the integers. On the recommendation of Vladimir Pestov (one of my advisors) I have been learning about T-sequences, which provide a rich method for producing topological groups with extreme behaviour. Here I will present two techniques involving T-sequences that help to answer two different questions about topological groups; one is about dynamics and the other is about combinatorial properties of . These results all come from “Topologies on the Integers” by Protasov and Zelenyuk.