A Topological Version for the Oligomorphicity of Groups

Given an action of a group on a set, the topology on the set determines the admissible group topologies on the group by which the group becomes a topological group and the action is continuous (and even yields uniformities on the set with respect to which the action extends continuously to the corresponding completions). This approach, which uses the discrete topology on the set and the permutation topology on the group, makes it possible to uncover connections between the oligomorphicity of the group action on the set, the total boundedness of the maximal equiuniformity on the set, and the Roelcke precompactness of the group. If the set is a simple chain then its ultrahomogeneity is equivalent to the oligomorphicity of the action of its automorphism group on the set, and is also equivalent to the Roelcke precompactness of the automorphism group itself, both in the permutation topology and in the topology of pointwise convergence.