{
  "id": "1710.06087",
  "version": "v1",
  "published": "2017-10-17T04:28:55.000Z",
  "updated": "2017-10-17T04:28:55.000Z",
  "title": "Small Cardinals and the Pseudocompactness of Hyperspaces of Subspaces of $βω$",
  "authors": [
    "Y. F. Ortiz-Castillo",
    "V. O. Rodrigues",
    "A. H. Tomita"
  ],
  "comment": "15 pages",
  "categories": [
    "math.GN"
  ],
  "abstract": "We study the relations between a generalization of pseudocompactness, named $(\\kappa, M)$-pseudocompactness, the countably compactness of subspaces of $\\beta \\omega$ and the pseudocompactness of their hyperspaces. We show, by assuming the existence of $\\mathfrak c$-many selective ultrafilters, that there exists a subspace of $\\beta \\omega$ that is $(\\kappa, \\omega^*)$-pseudocompact for all $\\kappa<\\mathfrak c$, but $\\text{CL}(X)$ isn't pseudocompact. We prove in ZFC that if $\\omega\\subseteq X\\subseteq \\beta\\omega$ is such that $X$ is $(\\mathfrak c, \\omega^*)$-pseudocompact, then $\\text{CL}(X)$ is pseudocompact, and we further explore this relation by replacing $\\mathfrak c$ for some small cardinals. We provide an example of a subspace of $\\beta \\omega$ for which all powers below $\\mathfrak h$ are countably compact whose hyperspace is not pseudocompact, we show that if $\\omega \\subseteq X$, the pseudocompactness of $\\text{CL}(X)$ implies that $X$ is $(\\kappa, \\omega^*)$-pseudocompact for all $\\kappa<\\mathfrak h$, and provide an example of such an $X$ that is not $(\\mathfrak b, \\omega^*)$-pseudocompact.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-17T04:28:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54B20",
      "03E17",
      "54D20"
    ],
    "keywords": [
      "small cardinals",
      "pseudocompactness",
      "hyperspace",
      "isnt pseudocompact",
      "countably compactness"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}