Topological field

A topological ring $K$ that is a field, where in addition it is required that the inverse mapping $a\mapsto a^{-1}$ is continuous on $K\setminus \{0\}$. Any subfield $P$ of a topological field $K$, and the closure $\overline{P}$ of $P$ in $K$, is also a topological field.

The only connected locally compact topological fields are $\mathbb R$ and $\mathbb C$ (cf. also Locally compact skew-field). Every normed field is a topological field with respect to the topology induced by the norm (cf. Norm; Norm on a field). If there exist two real-valued norms $u$ and $v$ on a field $P$, each of which makes $P$ a complete topological field, and if the topologies $\tau_u$ and $\tau_v$ induced by $u$ and $v$ are distinct, then the field $P$ is algebraically closed. The field $\mathbb C$ is the unique real-normed extension of the field $\mathbb R$.

On every field of infinite cardinality $\tau$ there exist precisely $2^{2^{\tau}}$ distinct topologies which make it a topological field. The topology of a topological field is either anti-discrete or completely regular. Topological fields with non-normal topologies, as well as topological fields with a normal but not hereditarily-normal topology, have been constructed. Topological fields are either connected or totally disconnected. There exists a connected topological field of arbitrary finite characteristic. It is unknown (1992) whether every topological field can be imbedded as a subfield in a connected topological field. In contrast to topological rings and linear topological spaces, not every completely-regular topological space can be imbedded as a subspace of a topological field. For example, a pseudo-compact (in particular, compact) subspace of a topological field is always metrizable. However, every completely-regular space admitting a continuous bijection onto a metric space can be imbedded as a subspace in some topological field. If the topological field $P$ contains at least one countable non-closed subset, then there exists a weakest metrizable topology on $P$ turning it into a topological field.

For a topological field $K$ one defines the completion $\tilde{K}$ — a complete topological ring in which $K$ is imbedded as an everywhere-dense subfield. The ring $\tilde{K}$ can have zero divisors. However, the completion of every real-normed topological field is a real-normed topological field.

References

[1]	L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian)
[2]	W. Wieslaw, "Topological fields" , M. Dekker (1988)
[3]	D.B. Shakhmatov, "Cardinal invariants of topological fields" Soviet. Math. Dokl. , 28 : 1 (1983) pp. 209–294 Dokl. Akad. Nauk SSSR , 271 : 6 (1983) pp. 1332–1336