Proposition 14
Tag 0C50, permanent. Cite it as [Bourbaki, Tag 0C50].
General Topology, chapter IX, § 3. Metrizable groups, valued fields, normed spaces and algebras
In a complete valued division ring, the multiplicative group of non-zero elements is a complete group.
The proof is similar to that of Proposition 13; we have only to replace the norm of by the absolute value of the division ring under consideration.
Note that we cannot apply Proposition 13 directly, because a (non-commutative) valued division ring is not necessarily an algebra over a non-discrete valued field (the restriction of the absolute value to the centre of the division ring might be improper).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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