The aim of this work is to show that every minimal action of integers on Cantor spaces is topologically conjugate to a Bratteli-Vershik map associated with some Bratteli diagram. We begin with the definitions of a Cantor space, of a partition, and of a refining sequence of partitions. We also introduce the notion of an ultrametric, and use it, together with a refining sequence of partitions, to characterise totally disconnected compact metric spaces. We further prove that all Cantor spaces are homeomorphic to one another. We then introduce the notions of topological conjugacy and minimal actions. Later, we define Bratteli diagrams and define a metric on the space of its infinite paths. We prove that this space of infinite paths is compact. We introduce the notion of a simple Bratteli diagram. We then order the edges of the diagram and, on ordered diagrams, define the Bratteli-Vershik map, whose minimality we characterise using simplicity. In the end, using Kakutani-Rokhlin partitions, we construct a Bratteli diagram for a minimal action on a Cantor space. The diagram and associated Bratteli-Vershik map are topologically conjugate to the Cantor space and minimal action.
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