{
  "id": "1205.6729",
  "version": "v1",
  "published": "2012-05-30T15:37:09.000Z",
  "updated": "2012-05-30T15:37:09.000Z",
  "title": "The partially ordered set of one-point extensions",
  "authors": [
    "M. R. Koushesh"
  ],
  "comment": "31 pages",
  "journal": "Topology Appl. 158 (2011), 509-532",
  "categories": [
    "math.GN"
  ],
  "abstract": "A space $Y$ is called an {\\em extension} of a space $X$ if $Y$ contains $X$ as a dense subspace. Two extensions of $X$ are said to be {\\em equivalent} if there is a homeomorphism between them which fixes $X$ point-wise. For two (equivalence classes of) extensions $Y$ and $Y'$ of $X$ let $Y\\leq Y'$ if there is a continuous function of $Y'$ into $Y$ which fixes $X$ point-wise. An extension $Y$ of $X$ is called a {\\em one-point extension} of $X$ if $Y\\backslash X$ is a singleton. Let ${\\mathcal P}$ be a topological property. An extension $Y$ of $X$ is called a {\\em ${\\mathcal P}$-extension} of $X$ if it has ${\\mathcal P}$. One-point ${\\mathcal P}$-extensions comprise the subject matter of this article. Here ${\\mathcal P}$ is subject to some mild requirements. We define an anti-order-isomorphism between the set of one-point Tychonoff extensions of a (Tychonoff) space $X$ (partially ordered by $\\leq$) and the set of compact non-empty subsets of its outgrowth $\\beta X\\backslash X$ (partially ordered by $\\subseteq$). This enables us to study the order-structure of various sets of one-point extensions of the space $X$ by relating them to the topologies of certain subspaces of its outgrowth. We conclude the article with the following conjecture. For a Tychonoff spaces $X$ denote by ${\\mathscr U}(X)$ the set of all zero-sets of $\\beta X$ which miss $X$. \\noindent{\\bf Conjecture.} {\\em For locally compact spaces $X$ and $Y$ the partially ordered sets $({\\mathscr U}(X),\\subseteq)$ and $({\\mathscr U}(Y),\\subseteq)$ are order-isomorphic if and only if the spaces ${\\em cl}_{\\beta X}(\\beta X\\backslash\\upsilon X)$ and ${\\em cl}_{\\beta Y}(\\beta Y\\backslash\\upsilon Y)$ are homeomorphic.}",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-05-30T15:37:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54D35",
      "54D40",
      "54D20"
    ],
    "keywords": [
      "partially ordered set",
      "one-point extension",
      "one-point tychonoff extensions",
      "compact non-empty subsets",
      "extensions comprise"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1205.6729K"
    }
  }
}