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A non-Archimedean normed field $K$ is said to be spherically complete if every decreasing sequence of closed balls in $K$ has non-empty intersection. I am a little puzzled as to why this definition is routinely restricted to the non-Archimedean case.

For example, in the Archimedean fields $\mathbb{R}$ and $\mathbb{C}$ with absolute value, it follows from Cantor's Intersection Theorem that every decreasing sequence of closed balls also has non-empty intersection. This fails for $\mathbb{Q}$, since $[\pi-1/n,\pi+1/n]\cap\mathbb{Q}$ is a decreasing sequence of closed balls in $\mathbb{Q}$ with empty intersection. So it cannot be true for all Archimedean normed fields.

Is there some crucial distinction between the Archimedean and non-Archimedean case vis-à-vis the spherical completeness property?

Many thanks

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  • $\begingroup$ One example comes from the theory of Berkovich spaces or Huber's adic space - such decreasing sequences of balls give rise to what is known as "type 4" points, which often have to be dealt wit separately. $\endgroup$
    – Wojowu
    Jun 20, 2021 at 21:32
  • $\begingroup$ Another example, though not one I can claim to understand well, is mentioned in Remark 4.2.5 of Bhatt's notes on perfectoids - apparently the field being spherically incomplete causes some Ext groups to be nontrivial. $\endgroup$
    – Wojowu
    Jun 20, 2021 at 21:34
  • $\begingroup$ @Wojowu Are you saying that these are examples of where it works very differently in the Archimedean case? I can't say I have any familiarity with the concepts you refer to, unfortunately. $\endgroup$ Jun 20, 2021 at 21:38
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    $\begingroup$ I think I might have slightly misread the intent of the question. I think the most fundamental answer to your question is that all complete fields are automatically spherically complete in the Archimedean world, but not in the non-Archimedean world. The examples I have mentioned are some of the subtleties which arise for complete, non-spherically complete fields. So the reason this notion is restricted to non-Archimedean world is because otherwise it is equivalent to regular completeness. $\endgroup$
    – Wojowu
    Jun 20, 2021 at 21:43

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