论文标题
拓扑先验领域
Topological Transcendental Fields
论文作者
论文摘要
本文启动了对所有复数数字的拓扑字段$ \ cc $的子场的拓扑先验领域$ \ ff $的研究，因此$ \ ff $仅由合理数字和一组非空的先验数字组成。 $ \ ff $，它以$ \ cc $的子空间继承的拓扑是一个拓扑字段。每个拓扑先验场都是一个可分离的零维空间，而代数为$ \ qq（t）$，这是理性数字的扩展，将有理数字段的扩展通过一组$ t $ t $ the Trabordental Numbers。事实证明，精确存在$ 2^{\ aleph_0} $可计数的无限拓扑超验域，并且每个字段都是同型与常规拓扑合理数字的空间$ \ qq $。还表明，有一类$ 2^{2^{\ Aleph_0}} $的$ \ qq（t）$的拓扑超验字段的$ 2^{\ aleph_0}} $，其中$ t $ a $ t $一组liouville编号，其中两个是同源的。
This article initiates the study of topological transcendental fields $\FF$ which are subfields of the topological field $\CC$ of all complex numbers such that $\FF$ consists of only rational numbers and a nonempty set of transcendental numbers. $\FF$, with the topology it inherits as a subspace of $\CC$, is a topological field. Each topological transcendental field is a separable metrizable zero-dimensional space and algebraically is $\QQ(T)$, the extension of the field of rational numbers by a set $T$ of transcendental numbers. It is proved that there exist precisely $2^{\aleph_0}$ countably infinite topological transcendental fields and each is homeomorphic to the space $\QQ$ of rational numbers with its usual topology. It is also shown that there is a class of $2^{2^{\aleph_0} }$ of topological transcendental fields of the form $\QQ(T)$ with $T$ a set of Liouville numbers, no two of which are homeomorphic.
