{
  "id": "1408.5015",
  "version": "v1",
  "published": "2014-08-21T14:21:12.000Z",
  "updated": "2014-08-21T14:21:12.000Z",
  "title": "Uniquely universal sets in $\\mathbb{R} \\times ω$ and $[0,1] \\times ω$",
  "authors": [
    "Alicja Krzeszowiec"
  ],
  "categories": [
    "math.GN"
  ],
  "abstract": "Let $X$ and $Y$ be topological spaces. We say that $X\\times Y$ satisfies the Uniquely Universal property (UU) iff there exists an open set $U\\subseteq X\\times Y$ such that for every open set $W\\subseteq Y$ there is a unique cross section of $U$ with $U\\left( x\\right) =\\left\\{ y\\in Y:\\left( x,y\\right) \\in U\\right\\} =W$. Arnold W. Miller in his paper \\cite{1} posed the following two questions: 1. Does $[ 0,1] \\times \\omega $ have UU? 2. Does $ \\mathbb{R} \\times \\omega $ have UU? In this paper we present two constructions which give positive answers to both problems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-21T14:21:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "uniquely universal sets",
      "open set",
      "unique cross section",
      "uniquely universal property",
      "topological spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.5015K"
    }
  }
}