{
  "id": "1602.04857",
  "version": "v1",
  "published": "2016-02-15T22:44:58.000Z",
  "updated": "2016-02-15T22:44:58.000Z",
  "title": "Fans and their applications in General Topology, Functional Analysis and Topological Algebra",
  "authors": [
    "Taras Banakh"
  ],
  "comment": "117 pages",
  "categories": [
    "math.GN",
    "math.CT"
  ],
  "abstract": "A family of closed subsets of a topological space $X$ is called a (strict) $Cld$-fan in $X$ if this family is (strictly) compact-finite but not locally finite in $X$. Applications of (strict) $Cld$-fans are based on a simple observation that $k$-spaces contain no $Cld$-fan and Ascoli spaces contain no strict $Cld$-fan. In this paper we develop the machinery of (strict) fans and apply it to detecting the $k$-space and Ascoli properties in spaces that naturally appear in General Topology, Functional Analysis, and Topological Algebra. In particular, we detect (generalized) metric spaces $X$ whose functor-spaces, functions spaces, free (para)topological (abelian) groups, free (locally convex) linear topological spaces, free (Lawson) topological semilattices, and free (para)topological (Clifford, Abelian) inverse semigroups are $k$-spaces or Ascoli spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-15T22:44:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54D50",
      "54E20",
      "54H10",
      "46B20"
    ],
    "keywords": [
      "functional analysis",
      "general topology",
      "topological algebra",
      "applications",
      "topological space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 117,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160204857B"
    }
  }
}