{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a set, let $Y$ be a [[topological space|0CQHFPGJ45]], and let $Y^X$ be the set of all maps from $X$ to $Y$.\nFor $x \\in X$ and an [[open|02BH3MCBQB]] set $U \\subseteq Y$, put\n$$W(x, U) = \\set{f \\in Y^X : f(x) \\in U}.$$\nThe *topology of pointwise convergence* on $Y^X$ is the topology generated by the [[subbasis|09S8DV34Y0]] of all sets $W(x, U)$.\nOn a subset $F \\subseteq Y^X$, such as the set of continuous maps when $X$ is a space, it is the [[subspace topology|0WS9P12FQR]].\n\n## Remarks\n\nViewing $Y^X$ as the product $\\prod_{x \\in X} Y$, this topology is exactly the [[product topology|0GBG8B49AA]].\nA sequence $(f_n)$ converges to $f$ in it if and only if $f_n(x) \\to f(x)$ in $Y$ for every $x \\in X$, which is [[pointwise convergence|0GFGK40582]].\nThe same holds for nets.\n\nExample: on $\\R^{[0, 1]}$, let $f_n(x) = x^n$, and let $f(x) = 0$ for $x \u003c 1$ and $f(1) = 1$.\nFor each fixed $x$, the numbers $x^n$ tend to $f(x)$, so $f_n \\to f$ in this topology.\n\nNon-example: the discrete topology on $\\R^{[0, 1]}$ is not the topology of pointwise convergence.\nIn the discrete topology, the sequence $f_n$ above does not converge, because $\\set{f}$ is open and contains no $f_n$.\n\nFor an uncountable set $X$ and $Y = \\R$, the space $\\R^X$ with this topology is not first-countable, so it is not metrizable.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "FVGB8ZP67B",
    "kind": "definition",
    "msc": [
      "54C35",
      "54B10"
    ],
    "name": "Topology of pointwise convergence",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Pointwise_convergence"
      }
    ],
    "tag": "D8GHJN61MW",
    "uses": [
      "0CQHFPGJ45",
      "0GBG8B49AA",
      "09S8DV34Y0",
      "02BH3MCBQB",
      "0GFGK40582",
      "0WS9P12FQR"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/D8GHJN61MW/",
  "links": [
    {
      "field": "defines",
      "target": "FVGB8ZP67B"
    },
    {
      "field": "uses",
      "target": "0CQHFPGJ45"
    },
    {
      "field": "uses",
      "target": "0GBG8B49AA"
    },
    {
      "field": "uses",
      "target": "09S8DV34Y0"
    },
    {
      "field": "uses",
      "target": "02BH3MCBQB"
    },
    {
      "field": "uses",
      "target": "0GFGK40582"
    },
    {
      "field": "uses",
      "target": "0WS9P12FQR"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/D8GHJN61MW.md",
  "msc": [
    "54C35",
    "54B10"
  ],
  "name": "Topology of pointwise convergence",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/D8G/D8GHJN61MW.md",
  "status": null,
  "tag": "D8GHJN61MW",
  "uri": "m2://D8GHJN61MW",
  "url": "https://math2.tamnd.com/wiki/topology-of-pointwise-convergence-definition/",
  "writing": {
    "tag": "D8GHJN61MW",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}