{
  "id": "2203.10725",
  "version": "v1",
  "published": "2022-03-21T04:02:28.000Z",
  "updated": "2022-03-21T04:02:28.000Z",
  "title": "Some properties of Pre-uniform spaces",
  "authors": [
    "Fucai Lin",
    "Yufan Xie",
    "Ting Wu",
    "Meng Bao"
  ],
  "comment": "17",
  "categories": [
    "math.GN"
  ],
  "abstract": "In this paper, we introduce the notions of pre-uniform spaces and pre-proximities and investigate some basic properties about them. First, we prove that each pre-uniform pre-topology is regular, and give an example to show that there exists a pre-uniform structure on a finite set such that the pre-uniform pre-topology is not discrete. Moreover, we give three methods of generating (strongly) pre-uniformities, that is, the definition of a pre-base, a family of strongly pre-uniform covers, or a family of strongly pre-uniform pseudometrics. As an application, we show that each strongly pre-topological group is completely regular. Finally, we pose the concept of the pre-proximity on a set and discuss some properties of the pre-proximity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-21T04:02:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54A05",
      "54C08",
      "54E05",
      "54E15"
    ],
    "keywords": [
      "pre-uniform spaces",
      "pre-uniform pre-topology",
      "pre-proximity",
      "basic properties",
      "pre-uniform structure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}