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Difference between revisions of "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.
 
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" />.
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The only connected locally compact topological fields are $\mathbb R$ and $\mathbb C$ (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 $u$ 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t09306022.png" /> there exist precisely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t09306023.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t09306024.png" /> contains at least one countable non-closed subset, then there exists a weakest metrizable topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t09306025.png" /> turning it into a topological field.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t09306026.png" /> one defines the completion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t09306027.png" /> — a complete topological ring in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t09306028.png" /> is imbedded as an everywhere-dense subfield. The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093060/t09306029.png" /> can have zero divisors. However, the completion of every real-normed topological field is a real-normed topological field.
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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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.S. Pontryagin,  "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Wieslaw,  "Topological fields" , M. Dekker  (1988)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.S. Pontryagin,  "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Wieslaw,  "Topological fields" , M. Dekker  (1988)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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</TD></TR></table>

Revision as of 08:25, 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 $\mathbb R$ 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
How to Cite This Entry:
Topological field. Encyclopedia of Mathematics. URL: http://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