A Uniqueness Result for the Group of Permutations of the Natural Numbers

Even in non-mathematical environments, a set is well known to be a collection of distinct, well-defined objects, named the elements of the set. A priori, a set can contain any number of elements. If an order can be defined on a set, then a permutation can be seen as any of the various ways in which its elements can be ordered. Now, how a permutation acts on a set can appear quite intuitive as we deal with a finite number of elements. Although the concept of permutation remains the same as we consider an infinite set, some aspects change radically and yield noteworthy results. The purpose of this work is to study the group of infinite permutations from a topological point of view and eventually prove a uniqueness result under defined hypotheses. More specifically: The combination of a group structure and a "compatible" topology yields what we call a topological group. Our interest is to study the case where a topological group also happens to be completely metrisable and separable, meaning that it has a compatible complete metric and a countable dense subset. When such conditions are fulfilled then a group is said to be Polish, and the relative topology is called a Polish group topology. In this paper we first go through some foundational theory and at the end we gather all previous knowledge to eventually prove that the group of permutations on the naturals admits a unique Polish group topology.