{
  "id": "2204.08703",
  "version": "v1",
  "published": "2022-04-19T07:09:04.000Z",
  "updated": "2022-04-19T07:09:04.000Z",
  "title": "On Function Spaces Related to H-sober Spaces",
  "authors": [
    "Meng Bao",
    "Xiaoyuan Zhang",
    "Xiaoquan Xu"
  ],
  "comment": "11 pages",
  "categories": [
    "math.GN"
  ],
  "abstract": "In this paper, we mainly study the function spaces related to H-sober spaces. For an irreducible subset system H and $T_{0}$ spaces $X$ and $Y$, it is proved that $Y$ is H-sober iff the function space $\\mathbb{C}(X, Y)$ of all continuous functions $f : X\\longrightarrow Y$ equipped with the topology of pointwise convergence is H-sober iff the function space $\\mathbb{C}(X, Y)$ equipped with the Isbell topology is H-sober. One immediate corollary is that for a $T_{0}$ space $X$, $Y$ is a sober space (resp., $d$-space, well-filtered space) iff the function space $\\mathbb{C}(X, Y)$ equipped with the topology of pointwise convergence is a sober space (resp., $d$-space, well-filtered space) iff the function space $\\mathbb{C}(X, Y)$ equipped with the the Isbell topology is a sober space (resp., $d$-space, well-filtered space). It is shown that $T_{0}$ spaces $X$ and $Y$, if the function space $\\mathbb{C}(X, Y)$ equipped with the compact-open topology is H-sober, then $Y$ is H-sober. The function space $\\mathbb{C}(X, Y)$ equipped with the Scott topology is also discussed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-19T07:09:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54C35",
      "54D99",
      "06B30",
      "06B35"
    ],
    "keywords": [
      "function space",
      "h-sober spaces",
      "well-filtered space",
      "isbell topology",
      "pointwise convergence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}