{
  "id": "1908.02845",
  "version": "v1",
  "published": "2019-08-07T21:23:58.000Z",
  "updated": "2019-08-07T21:23:58.000Z",
  "title": "Hyperspaces of infinite compacta with finitely many accumulation points",
  "authors": [
    "Paweł Krupski",
    "Krzysztof Omiljanowski"
  ],
  "comment": "17 pages",
  "categories": [
    "math.GN"
  ],
  "abstract": "Vietoris hyperspaces $\\mathcal A_n(X)$ ($\\mathcal A_\\omega(X)$) of infinite compact subsets of a metric space $X$ which have at most $n$ (finitely many, resp.) accumulation points are studied. If $X$ is a dense-in-itself, 0-dimensional Polish space, then $\\mathcal A_n(X)$ is homeomorphic to the product $\\mathbb Q^{\\mathbb N}$. The hyperspaces $\\mathcal A_1(\\mathbb Q,\\{q\\})$ and $\\mathcal A_1(\\mathbb R\\setminus\\mathbb Q,\\{p\\})$ of all $A\\in \\mathcal A_1(\\mathbb Q)$ (resp. $A\\in \\mathcal A_1(\\mathbb R\\setminus\\mathbb Q)$) which accumulate at $q\\in\\mathbb Q$ (resp. $p\\in \\mathbb R\\setminus\\mathbb Q$) are also homeomorphic to $\\mathbb Q^{\\mathbb N}$. If $X$ is a nondegenerate locally connected metric continuum then hyperspaces $\\mathcal A_n(X)$ are absolute retracts for all $n\\in\\mathbb N\\cup\\{\\omega\\}$. If $X=J=[-1,1]$ or $X=S^1$, the hyperspaces $\\mathcal A_n(X)$ are characterized as $F_{\\sigma\\delta}$-absorbers in hyperspaces $\\mathcal K(J)$ and $\\mathcal K(S^1)$ of all compacta in $J$ and $S^1$, respectively. Consequently, they are homeomorphic to the linear space $\\{(x_k) \\in\\mathbb R^{\\mathbb N}: \\lim x_k=0\\}$ with the product topology. The hyperspaces $\\mathcal A_\\omega(X)$ for $X$ being a Euclidean cube, the Hilbert cube, the $m$-dimensional unit sphere $S^m$, $m\\ge 1$, or a compact $m$-manifold with boundary in $S^m$, $m\\ge 3$, are true $F_{\\sigma\\delta\\sigma}$-sets which are strongly $F_{\\sigma\\delta}$-universal in the respective hyperspaces of all compacta.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-07T21:23:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57N20",
      "54B20",
      "54H05"
    ],
    "keywords": [
      "hyperspaces",
      "accumulation points",
      "infinite compacta",
      "dimensional unit sphere",
      "infinite compact subsets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}