$p$ -自闭域上可解释群的拓扑化
Topologizing interpretable groups in $p$-adically closed fields
摘要：
本文研究了$p$ -根闭域$K$中的可解拓扑空间和拓扑群.我们确定了一类特殊的“容许拓扑”的拓扑驯服属性，如一般的连续性，类似于拓扑上的可定义子集的$K^n$。证明了每个可解释集合至少有一个容许拓扑，每个可解释群有唯一的容许群拓扑。然后，我们考虑可解释群上的可定义紧性（在Fornasiero意义上）。我们证明了一个可解释群是可定义紧的当且仅当它具有可满足的泛类，推广了可定义群的一个较早的结果。因此，我们看到fsg是可解释群的可定义族中的可定义性质，并且任何定义在$\mathbb{Q}_p$上的fsg可解释群都可定义同构于可定义群。
英文摘要：
We consider interpretable topological spaces and topological groups in a $p$-adically closed field $K$. We identify a special class of "admissible topologies" with topological tameness properties like generic continuity, similar to the topology on definable subsets of $K^n$. We show every interpretable set has at least one admissible topology, and every interpretable group has a unique admissible group topology. We then consider definable compactness (in the sense of Fornasiero) on interpretable groups. We show that an interpretable group is definably compact if and only if it has finitely satisfiable generics (fsg), generalizing an earlier result on definable groups. As a consequence, we see that fsg is a definable property in definable families of interpretable groups, and that any fsg interpretable group defined over $\mathbb{Q}_p$ is definably isomorphic to a definable group.
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