{
  "id": "2403.04004",
  "version": "v1",
  "published": "2024-03-06T19:28:41.000Z",
  "updated": "2024-03-06T19:28:41.000Z",
  "title": "Approximation by continuous functions and its applications",
  "authors": [
    "Anton E. Lipin",
    "Alexander V. Osipov"
  ],
  "comment": "15 pages",
  "categories": [
    "math.GN",
    "math.CA"
  ],
  "abstract": "We prove that for every normal topological space $X$ and any function $f: X \\to \\mathbb{R}$ there is a continuous function $g : X \\to \\mathbb{R}$ such that $$|f(x) - g(x)| \\leq \\frac{1}{2} \\sup\\limits_{p \\in X} \\inf\\limits_{O(p)} \\sup\\limits_{a,b \\in O(p)} |f(a) - f(b)|$$ for all $x \\in X$. As an application of this result we prove the following statements to types of tightness in a space $Q_p(X, \\mathbb{R})$ of all quasicontinuous real-valued functions with the topology $\\tau_p$ of pointwise convergence: the countability of tightness (fan-tightness, strong fan-tightness) at a point $f$ of space $Q_p(X, \\mathbb{R})$ implies the countability of tightness (fan-tightness, strong fan-tightness) of space $Q_p(X,Y)$ of all quasicontinuous functions from $X$ into any non-one-point metrizable space $Y$. This result is the answer to the open question in the class of metrizable spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-06T19:28:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "application",
      "approximation",
      "strong fan-tightness",
      "countability",
      "normal topological space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}