Talk:Spherically complete field
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open balls?
[edit]The topological field of real numbers is locally compact, but the decreasing sequence of balls given by the open intervals (0, 1/n) has empty intersection.
Therefore, while in the non-archimedean case, open and closed balls can be interchangeably used, is this still true in the archimedean case, or might it be necessary to stipulate open balls? — Preceding unsigned comment added by 179.235.134.104 (talk) 23:57, 19 March 2019 (UTC)
- Unless I'm wrong, on the contrary, this example shows that one should stipulate closed balls! (As you say, R is spherically complete since locally compact.) — MFH:Talk 16:39, 20 June 2019 (UTC)
why only fields?
[edit]Why has Wikipedia "spherically complete" only for fields? This notion makes sense in any metric space, doesn't it? — MFH:Talk 16:42, 20 June 2019 (UTC)
Can you create a spherical completion of any normed field?
[edit]The article does not mention circumstances when one may create a spherical completion of a field that is not already spherically complete.
That would be a very useful addition to this article.
Some Ideas for the article
[edit]Somethings one could consider adding:
1. For archimedean fields being spherical complete is equivalent to the usual notion of completeness. Thats why it is typically only considered for nonarchimedean fields.
2. Spherical completeness is equivalent to being maximally complete i.e. the field has no nontrivial immediate field extension. A field extension is called immediate if both fields have the same value group and residue field. See Irving Kaplansky "Maximal fields with valuations," Duke Mathematical Journal, Duke Math. J. 9(2), 303-321, (June 1942)
3. As someone else already asked about: Every valued field has a spherical completion. This completion can be constructed using Hahn series and is unique for characteristic 0 but not necessarily in the characterstic p case. See Poonen, Bjorn. "Maximally complete fields." Enseign. Math 39.1-2 (1993): 87-106.
When going through older sources you may find that they talk about pseudo convergent sequences and pseudo limits rather than spherical completeness. However its not hard to see that these two are the same concept just phrased differently.