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<p>Given an action of a group on a set, the topology on the set determines the admissible grouptopologies on the group by which the group becomes a topological groupand the action is&#xa0;continuous (and even yields uniformities on the set with respect to whichthe 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 equivalentto 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.</p>

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A Topological Version for the Oligomorphicity of Groups

  • 期刊论文

  • 发表日期：

    2025年5月1日

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  • B. V. Sorin

摘要

Given an action of a group on a set, the topology on the set determines the admissible grouptopologies on the group by which the group becomes a topological groupand the action is continuous (and even yields uniformities on the set with respect to whichthe 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 equivalentto 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.