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Feb 15, 2022 at 9:06 comment added darij grinberg There are two German papers by Andreas Golembiowski that appear to address this question (I have not read them myself).
Feb 15, 2022 at 8:50 history edited YCor CC BY-SA 4.0
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Feb 15, 2022 at 8:44 answer added Christopher Ryba timeline score: 16
Feb 15, 2022 at 8:37 history edited YCor CC BY-SA 4.0
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Feb 15, 2022 at 8:36 comment added YCor As regards the whole symmetric group $S_\omega$ (I loathe the notation $S_\infty$, which implicitly suggests that any infinite set is countable — the symmetric groups of uncountable sets are interesting too), this is naturally a topological group under the Onofri topology (of pointwise convergence). When considering modules over it, it is therefore natural to expect continuity of the action and possibly topological modules.
Feb 15, 2022 at 8:34 comment added YCor The discussion is likely to be confusing if one uses the same notation for the whole symmetric group and its finitary subgroup...
Feb 15, 2022 at 6:05 history asked GA316 CC BY-SA 4.0