{
  "id": "2108.01195",
  "version": "v1",
  "published": "2021-08-02T22:19:03.000Z",
  "updated": "2021-08-02T22:19:03.000Z",
  "title": "$k$-spaces, sequential spaces and related topics in the absence of the axiom of choice",
  "authors": [
    "Kyriakos Keremedis",
    "Eliza Wajch"
  ],
  "categories": [
    "math.GN"
  ],
  "abstract": "In the absence of the axiom of choice, new results concerning sequential, Fr\\'echet-Urysohn, $k$-spaces, very $k$-spaces, Loeb and Cantor completely metrizable spaces are shown. New choice principles are introduced. Among many other theorems, it is proved in $\\mathbf{ZF}$ that every Loeb, $T_3$-space having a base expressible as a countable union of finite sets is a metrizable second-countable space whose every $F_{\\sigma}$-subspace is separable; moreover, every $G_{\\delta}$-subspace of a second-countable, Cantor completely metrizable space is Cantor completely metrizable, Loeb and separable. It is also noticed that Arkhangel'skii's statement that every very $k$-space is Fr\\'echet-Urysohn is unprovable in $\\mathbf{ZF}$ but it holds in $\\mathbf{ZF}$ that every first-countable, regular very $k$-space whose family of all non-empty compact sets has a choice function is Fr\\'echet-Urysohn. That every second-countable metrizable space is a very $k$-space is equivalent to the axiom of countable choice for $\\mathbb{R}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-02T22:19:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E25",
      "03E35",
      "54D50",
      "54D55",
      "54E50"
    ],
    "keywords": [
      "sequential spaces",
      "related topics",
      "metrizable space",
      "non-empty compact sets",
      "frechet-urysohn"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}