{
  "id": "2305.07354",
  "version": "v1",
  "published": "2023-05-12T10:05:54.000Z",
  "updated": "2023-05-12T10:05:54.000Z",
  "title": "Banakh spaces and their geometry",
  "authors": [
    "Taras Banakh"
  ],
  "comment": "21 pages",
  "categories": [
    "math.GN",
    "math.MG"
  ],
  "abstract": "Following a suggestion of WIll Brian, we define a metric space $(X,d)$ to be $Banakh$ if for every point $c\\in X$ and real number $r\\in d[X^2]$ the sphere $S(c;r):=\\{x\\in X:d(x,c)=r\\}$ is equal to $\\{x,y\\}$ for some points $x,y\\in X$ with $d(x,y)=2r$. We prove that for any points $x,y\\in X$ of a Banakh space with $r:=d(x,y)$, there exists a unique isometric embedding $\\ell_{xy}:r\\mathbb Z\\to X$ such that $\\ell_{xy}(0)=x$ and $\\ell_{xy}(r)=y$. Using this embedding result, we prove that a metric space $X$ is isometric to a subgroup $G$ of the additive group $\\mathbb Q$ of rational numbers if and only if $X$ is a Banakh space such that $d[X^2]=G_+:=\\{x\\in G:x\\ge 0\\}$. Also we prove that a metric space $X$ is isometric to the real line if and only if $X$ is a complete Banakh space such that $G_+\\subseteq d[X^2]$ for some non-cyclic subgroup $G\\subseteq\\mathbb Q$. We prove that every Banakh space $X$ with $d[X^2]\\subseteq\\mathbb Q$ is isometric to some subgroup of $\\mathbb Q$. Every Banakh space $X$ satisfies the ``rational'' axiom of segment construction: for every $x,y\\in X$ and every $r\\in\\mathbb Q{\\cdot} d(x,y)$ there exists a unique point $z\\in X$ such that $d(y,z)=r$ and $d(x,z)=d(x,y)+d(y,z)$. Given a subgroup $G$ of $\\mathbb Q$, we show that for every $c\\in X$ the $G$-sphere $S(c;G):=\\{x\\in X:d(c,x)\\in G\\}$ is isometric to a subgroup of $G$ if and only if the metric space $S(x;G)$ is centrally symmetric in the sense that for any distinct points $a,b\\in S(x;G)$ there exists an isometry $f$ of $S(c;G)$ such that $f(a)=a$ and $f(b)\\ne b$. Finally, for every nonzero cardinal $\\kappa\\le\\mathfrak c$ in the Hilbert space $\\ell_2(\\kappa)$ we construct a discrete subgroup $H$ which is a complete Banakh space of cardinality $|H|=\\max\\{\\kappa,\\omega\\}$, and a dense $\\mathbb Q$-linear subspace $L$ in $\\ell_2(\\kappa)$ such that $L$ is a Banakh space with $d[L^2]=\\mathbb R_+$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-12T10:05:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30L05",
      "46B85",
      "51F20",
      "54C35",
      "54E50"
    ],
    "keywords": [
      "metric space",
      "complete banakh space",
      "non-cyclic subgroup",
      "unique isometric",
      "rational numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}