{
  "id": "2402.11213",
  "version": "v1",
  "published": "2024-02-17T07:46:04.000Z",
  "updated": "2024-02-17T07:46:04.000Z",
  "title": "On resolvability and tightness in uncountable spaces",
  "authors": [
    "Anton Lipin"
  ],
  "comment": "12 pages",
  "categories": [
    "math.GN"
  ],
  "abstract": "We investigate connections between resolvability and different forms of tightness. This study is adjacent to [1,2]. We construct a non-regular refinement $\\tau^*$ of the natural topology of the real line $\\mathbb{R}$ with properties such that the space $(\\mathbb{R}, \\tau^*)$ has a hereditary nowhere dense tightness and it has no $\\omega_1$-resolvable subspaces, whereas $\\Delta(\\mathbb{R}, \\tau^*) = \\frak{c}$. We also show that the proof of the main result of [1], being slightly modified, leads to the following strengthening: if $L$ is a Hausdorff space of countable character and the space $L^\\omega$ is c.c.c., then every submaximal dense subspace of $L^\\kappa$ has disjoint tightness. As a corollary, for every $\\kappa \\geq \\omega$ there is a Tychonoff submaximal space $X$ such that $|X|=\\Delta(X)=\\kappa$ and $X$ has disjoint tightness.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-02-17T07:46:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "uncountable spaces",
      "resolvability",
      "disjoint tightness",
      "tychonoff submaximal space",
      "submaximal dense subspace"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}