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Topological Transcendental Fields

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journal contribution
posted on 04.05.2022, 04:30 authored by TP Chalebgwa, Sidney MorrisSidney Morris
This article initiates the study of topological transcendental fields F which are subfields of the topological field C of all complex numbers such that F only consists of rational numbers and a nonempty set of transcendental numbers. F, with the topology it inherits as a subspace of C, is a topological field. Each topological transcendental field is a separable metrizable zero-dimensional space and algebraically is Q(T), the extension of the field of rational numbers by a set T of transcendental numbers. It is proven that there exist precisely 2ℵ0 countably infinite topological transcendental fields and each is homeomorphic to the space Q of rational numbers with its usual topology. It is also shown that there is a class of 22ℵ0 of topological transcendental fields of the form Q(T) with T a set of Liouville numbers, no two of which are homeomorphic.

Funding

The first author's research is supported by the Fields Institute for Research in Mathematical Sciences, via the Fields-Ontario postdoctoral Fellowship.

History

Publication Date

01/01/2022

Journal

Axioms

Volume

11

Issue

3

Article Number

ARTN 118

Pagination

5p.

Publisher

Multidisciplinary Digital Publishing Institute (MDPI)

ISSN

2075-1680

Rights Statement

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/)