{
  "id": "1212.5749",
  "version": "v3",
  "published": "2012-12-23T01:08:53.000Z",
  "updated": "2013-11-14T01:58:54.000Z",
  "title": "Free paratopological groups",
  "authors": [
    "Ali Sayed Elfard"
  ],
  "categories": [
    "math.GN"
  ],
  "abstract": "Let $\\FP(X)$ be the free paratopological group on a topological space $X$ in the sense of Markov. In this paper, we study the group $\\FP(X)$ on a $P_\\alpha$-space $X$ where $\\alpha$ is an infinite cardinal and then we prove that the group $\\FP(X)$ is an Alexandroff space if $X$ is an Alexandroff space. Moreover, we introduce a neighborhood base at the identity of the group $\\FP(X)$ when the space $X$ is Alexandroff and then we give some properties of this neighborhood base. As applications of these, we prove that the group $\\FP(X)$ is $T_0$ if $X$ is $T_0$, we characterize the spaces $X$ for which the group $\\FP(X)$ is a topological group and then we give a class of spaces $X$ for which the group $\\FP(X)$ has the inductive limit property.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-11-14T01:58:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "free paratopological group",
      "neighborhood base",
      "alexandroff space",
      "infinite cardinal",
      "inductive limit property"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1212.5749S"
    }
  }
}