{
  "id": "1611.08289",
  "version": "v1",
  "published": "2016-11-24T19:18:16.000Z",
  "updated": "2016-11-24T19:18:16.000Z",
  "title": "Categorical properties on the hyperspace of nontrivial convergent sequences",
  "authors": [
    "S. Garcia-Ferreira",
    "R. Rojas-Hernandez",
    "Y. F. Ortiz-Castillo"
  ],
  "categories": [
    "math.GN"
  ],
  "abstract": "In this paper, we shall study categorial properties of the hyperspace of all nontrivial convergent sequences $\\mathcal{S}_c(X)$ of a Fre\\'ech-Urysohn space $X$, this hyperspace is equipped with the Vietoris topology. We mainly prove that $\\mathcal{S}_c(X)$ is meager whenever $X$ is a crowded space, as a corollary we obtain that if $\\mathcal{S}_c(X)$ is Baire, the $X$ has a dense subset of isolated points. As an interesting example $\\mathcal{S}_c(\\omega_1)$ has the Baire property, where $\\omega_1$ carries the order topology (this answers a question from \\cite{sal-yas}). We can give more examples like this one by proving that the Alexandroff duplicated $\\mathcal{A}(Z)$ of a space $Z$ satisfies that $\\mathcal{S}_c(\\mathcal{A}(Z))$ has the Baire property, whenever $Z$ is a $\\Sigma$-product of completely metrizable spaces and $Z$ is crowded. Also we show that if $\\mathcal{S}_c(X)$ is pseudocompact, then $X$ has a relatively countably compact dense subset of isolated points, every finite power of $X$ is pseudocompact, and every $G_\\delta$-point in $X$ must be isolated. We also establish several topological properties of the hyperspace of nontrivial convergent sequences of countable Fre\\'ech-Urysohn spaces with only one non-isolated point.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-24T19:18:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nontrivial convergent sequences",
      "categorical properties",
      "hyperspace",
      "freech-urysohn space",
      "baire property"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}