{
  "id": "1909.07693",
  "version": "v1",
  "published": "2019-09-17T10:11:34.000Z",
  "updated": "2019-09-17T10:11:34.000Z",
  "title": "Metrizability of $b$-metric space and $θ$-metric space via Chittenden's metrization theorem",
  "authors": [
    "Sumit Som"
  ],
  "categories": [
    "math.GN"
  ],
  "abstract": "In [An, V.T., Tuyen, Q.L., Dung, V.N., Stone-type theorem on $b$-metric spaces and applications, Topology Appl. 185-186 (2015) 50-64], Tran Van An et al. provide a sufficient condition for $b$-metric space to be metrizable. They proved the metrizability by assuming that the distance function is continuous in one variable. The main purpose of this manuscript is to provide a direct short proof of the metrizability of $b$-metric space introduced by Khamsi and Hussain in \\cite[\\, Khamsi, M.A and Hussain, N., KKM mappings in metric type spaces, Nonlinear Anal. 73 (9) (2010) 3123-3129]{kh} via Chittenden's metrization theorem without any assumption on the distance function. Further in this short note, we prove the metrizability of $\\theta$-metric space introduced by Khojasteh et al. in [Khojasteh, F., Karapinar, E., Radenovic, S., $\\theta$-metric space: A Generalization, Mathematical problems in Engineering, Volume 2013, Article 504609, 7 pages].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-17T10:11:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "metric space",
      "chittendens metrization theorem",
      "metrizability",
      "distance function",
      "direct short proof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}