{
  "id": "2307.07519",
  "version": "v1",
  "published": "2023-07-03T09:02:15.000Z",
  "updated": "2023-07-03T09:02:15.000Z",
  "title": "More on generalizations of topology of uniform convergence and $m$-topology on $C(X)$",
  "authors": [
    "Pratip Nandi",
    "Rakesh Bharati",
    "Atasi Deb Ray",
    "Sudip Kumar Acharyya"
  ],
  "categories": [
    "math.GN"
  ],
  "abstract": "This paper conglomerates our findings on the space $C(X)$ of all real valued continuous functions, under different generalizations of the topology of uniform convergence and the $m$-topology. The paper begins with answering all the questions which were left open in our previous paper on the classifications of $Z$-ideals of $C(X)$ induced by the $U_I$ and the $m_I$-topologies on $C(X)$. Motivated by the definition of $m^I$-topology, another generalization of the topology of uniform convergence, called $U^I$-topology, is introduced here. Among several other results, it is established that for a convex ideal $I$, a necessary and sufficient condition for $U^I$-topology to coincide with $m^I$-topology is the boundedness of $X\\setminus\\bigcap Z[I]$ in $X$. As opposed to the case of the $U_I$-topologies (and $m_I$-topologies), it is proved that each $U^I$-topology (respectively, $m^I$-topology) on $C(X)$ is uniquely determined by the ideal $I$. In the last section, the denseness of the set of units of $C(X)$ in $C_U(X)$ (= $C(X)$ with the topology of uniform convergence) is shown to be equivalent to the strong zero dimensionality of the space $X$. Also, the space $X$ is a weakly P-space if and only if the set of zero divisors (including 0) in $C(X)$ is closed in $C_U(X)$. Computing the closure of $C_\\mathscr{P}(X)$ (=$\\{f\\in C(X):\\text{the support of }f\\in\\mathscr{P}\\}$ where $\\mathscr{P}$ denotes the ideal of closed sets in $X$) in $C_U(X)$ and $C_m(X)$ (= $C(X)$ with the $m$-topology), the results $cl_UC_\\mathscr{P}(X) = C_\\infty^\\mathscr{P}(X)$ ($=\\{f\\in C(X):\\forall n\\in\\mathbb{N}, \\{x\\in X:|f(x)|\\geq\\frac{1}{n}\\}\\in\\mathscr{P}\\}$) and $cl_mC_\\mathscr{P}(X)=\\{f\\in C(X):f.g\\in C^\\mathscr{P}_\\infty(X)\\text{ for each }g\\in C(X)\\}$ are achieved.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-03T09:02:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "uniform convergence",
      "generalization",
      "strong zero dimensionality",
      "convex ideal",
      "paper conglomerates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}