# 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 | + | 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 | + | 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 | + | 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