{
  "id": "2202.00837",
  "version": "v1",
  "published": "2022-02-02T01:18:56.000Z",
  "updated": "2022-02-02T01:18:56.000Z",
  "title": "Topological Transcendental Fields",
  "authors": [
    "Taboka Prince Chalebgwa",
    "Sidney A. Morris"
  ],
  "comment": "6 pages",
  "categories": [
    "math.GN"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-02T01:18:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rational numbers",
      "transcendental numbers",
      "topological field",
      "countably infinite topological transcendental fields",
      "separable metrizable zero-dimensional space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}