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A [[Topological ring|topological ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t0930601.png" /> that is a [[Field|field]], where in addition it is required that the inverse mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t0930602.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t0930603.png" />. Any subfield <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t0930604.png" /> of a topological field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t0930605.png" />, and the closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t0930606.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t0930607.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t0930608.png" />, is also a topological field.
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A [[Topological ring|topological ring]] $K$ that is a [[Field|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t0930609.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t09306010.png" /> (cf. also [[Locally compact skew-field|Locally compact skew-field]]). Every normed field is a topological field with respect to the topology induced by the norm (cf. [[Norm|Norm]]; [[Norm on a field|Norm on a field]]). If there exist two real-valued norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t09306011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t09306012.png" /> on a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t09306013.png" />, each of which makes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t09306014.png" /> a complete topological field, and if the topologies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t09306015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t09306016.png" /> induced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t09306017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t09306018.png" /> are distinct, then the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t09306019.png" /> is algebraically closed. The field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t09306020.png" /> is the unique real-normed extension of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t09306021.png" />.
 
The only connected locally compact topological fields are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t0930609.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t09306010.png" /> (cf. also [[Locally compact skew-field|Locally compact skew-field]]). Every normed field is a topological field with respect to the topology induced by the norm (cf. [[Norm|Norm]]; [[Norm on a field|Norm on a field]]). If there exist two real-valued norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t09306011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t09306012.png" /> on a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t09306013.png" />, each of which makes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t09306014.png" /> a complete topological field, and if the topologies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t09306015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t09306016.png" /> induced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t09306017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t09306018.png" /> are distinct, then the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t09306019.png" /> is algebraically closed. The field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t09306020.png" /> is the unique real-normed extension of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t09306021.png" />.

Revision as of 05:52, 24 January 2013

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 and (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 and on a field , each of which makes a complete topological field, and if the topologies and induced by and are distinct, then the field is algebraically closed. The field is the unique real-normed extension of the field .

On every field of infinite cardinality there exist precisely 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 contains at least one countable non-closed subset, then there exists a weakest metrizable topology on turning it into a topological field.

For a topological field one defines the completion — a complete topological ring in which is imbedded as an everywhere-dense subfield. The ring 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
How to Cite This Entry:
Topological field. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Topological_field&oldid=29340
This article was adapted from an original article by D.B. Shakhmatov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article