{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a set and let $\\Delta = \\set{(x, x) : x \\in X}$ be the diagonal of $X \\times X$.\nFor relations $U, V \\subseteq X \\times X$, write $U^{-1} = \\set{(y, x) : (x, y) \\in U}$ and $V \\circ U = \\set{(x, z) : \\text{there is } y \\text{ with } (x, y) \\in U \\text{ and } (y, z) \\in V}$.\nA *uniformity* on $X$ is a nonempty family $\\mathcal{U}$ of subsets of $X \\times X$ such that:\n\n1. Every $U \\in \\mathcal{U}$ contains $\\Delta$.\n2. If $U \\in \\mathcal{U}$ and $U \\subseteq V \\subseteq X \\times X$, then $V \\in \\mathcal{U}$.\n3. If $U, V \\in \\mathcal{U}$, then $U \\cap V \\in \\mathcal{U}$.\n4. If $U \\in \\mathcal{U}$, then there is $V \\in \\mathcal{U}$ with $V \\circ V \\subseteq U$.\n5. If $U \\in \\mathcal{U}$, then $U^{-1} \\in \\mathcal{U}$.\n\nA *uniform space* is a pair $(X, \\mathcal{U})$ of a set $X$ and a uniformity $\\mathcal{U}$ on $X$.\n\n## Remarks\n\nLet $(X, d)$ be a [[metric space|0VKD1TP7F3]] and let $U_\\varepsilon = \\set{(x, y) : d(x, y) \u003c \\varepsilon}$.\nThe sets that contain some $U_\\varepsilon$ with $\\varepsilon \u003e 0$ form a uniformity.\nClauses 1, 2, 3 and 5 follow from $d(x, x) = 0$, from $U_\\varepsilon \\cap U_\\delta = U_{\\min(\\varepsilon, \\delta)}$ and from symmetry of $d$.\nClause 4 holds with $V = U_{\\varepsilon/2}$, by the [[triangle inequality|01M6CW9VM6]].\nOn $\\R$, the family of sets that contain $\\set{(x, y) : \\abs{x - y} \u003c 1}$ satisfies clauses 1, 2, 3 and 5 but not 4.\nFor $V$ in this family, $V \\circ V$ contains $\\set{(x, y) : \\abs{x - y} \u003c 2}$, which does not lie in $\\set{(x, y) : \\abs{x - y} \u003c 1}$.\nThe members of $\\mathcal{U}$ are its [[entourages|FVFFKNKEDF]].\nThe uniform topology on $X$ declares $O \\subseteq X$ open when for each $x \\in O$ there is $U \\in \\mathcal{U}$ with $\\set{y : (x, y) \\in U} \\subseteq O$.\n\n## Explanations\n\n### L1 human\n\nA uniformity is a list of \"closeness relations\" on a set.\nEach relation is a set of pairs of points that count as close.\nThe clauses say that every point is close to itself and that closeness relations can be enlarged and combined.\nEach relation can be turned around, and inside each relation lies a smaller one that, applied twice, stays inside the first, like $\\varepsilon/2$ and $\\varepsilon$ in a metric space.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "A54T86KS63",
    "kind": "definition",
    "msc": [
      "54E15"
    ],
    "name": "Uniform space",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Uniform_space"
      }
    ],
    "tag": "YWW42G978D",
    "uses": [
      "09CEDJYBKP",
      "05AVARPXA2",
      "0PYSR20HGF"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/YWW42G978D/",
  "links": [
    {
      "field": "defines",
      "target": "A54T86KS63"
    },
    {
      "field": "uses",
      "target": "09CEDJYBKP"
    },
    {
      "field": "uses",
      "target": "05AVARPXA2"
    },
    {
      "field": "uses",
      "target": "0PYSR20HGF"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/YWW42G978D.md",
  "msc": [
    "54E15"
  ],
  "name": "Uniform space",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/YWW/YWW42G978D.md",
  "status": null,
  "tag": "YWW42G978D",
  "uri": "m2://YWW42G978D",
  "url": "https://math2.tamnd.com/wiki/uniform-space-definition/",
  "writing": {
    "tag": "YWW42G978D",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}