Proposition 7

The completion $\hat{K}$ of a Hausdorff topological division ring K is a topological division ring if and only if the image under the mapping $x \to x^{-1}$ of every Cauchy filter (with respect to the additive structure) which does not have a cluster point at 0, is a Cauchy filter (with respect to the additive structure).

There are topological division rings in which this condition is not satisfied and in which the ring $\hat{K}$ has zero divisors (see Exercise 26). Moreover, even if the completion $\hat{K}$ is a topological division ring, there is no a priori reason to assume that the multiplicative structures of $\hat{K}$ are structures of a complete space. However, this will be the case for division rings K such that $\hat{K}$ is locally compact (see Chapter I, § 9, no. 7, Proposition 13, and Chapter III, § 3, no. 3, Proposition 4) and for topological fields; for the latter, we have the following proposition: