{
  "id": "1809.03085",
  "version": "v1",
  "published": "2018-09-10T02:11:16.000Z",
  "updated": "2018-09-10T02:11:16.000Z",
  "title": "Connected door spaces and topological solutions of equations",
  "authors": [
    "Jianfeng Wu",
    "Chunli Wang",
    "Dong Zhang"
  ],
  "comment": "A small gap in the proof of Theorem 3 in the published version has been corrected",
  "doi": "10.1007/s00010-018-0577-0",
  "categories": [
    "math.GN"
  ],
  "abstract": "The connected door space is an enigmatic topological space in which every proper nonempty subset is either open or closed, but not both. This paper provides an elementary proof of the classification theorem of connected door spaces. More importantly, we show that connected door topologies can be viewed as solutions of the valuation $f(A)+f(B)=f(A\\cup B)+f(A\\cap B)$ and the equation $f(A)+f(B)=f(A\\cup B)$, respectively. In addition, some special solutions, which can be regarded as a union of connected door spaces, are provided.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-10T02:11:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "39B05",
      "39B52",
      "54H99"
    ],
    "keywords": [
      "connected door space",
      "topological solutions",
      "proper nonempty subset",
      "elementary proof",
      "classification theorem"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}