{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "9VY90HJVZX"
    },
    {
      "field": "uses",
      "target": "FFH90QRFFJ"
    },
    {
      "field": "uses",
      "target": "K99YMX66XD"
    },
    {
      "field": "uses",
      "target": "YCQFP0KN48"
    }
  ],
  "body": "\n## Description\n\nLet $n \\ge 1$ be an integer.\nThe *real projective space* $\\mathbb{RP}^n$ is the set of lines through the origin in $\\R^{n+1}$.\nAs a topological space it is the [[quotient|0NQ1H2X495]] of the [[$n$-sphere|T62369RFH6]] $S^n$ by the relation that identifies each point $x$ with its antipode $-x$.\nIt is a compact [[topological manifold|QQQ9B1DK4P]] of dimension $n$.\nFor $i = 1, \\ldots, n+1$, let $U_i$ be the image of $\\set{x \\in S^n : x_i \\ne 0}$.\nThe map $U_i \\to \\R^n$ sending the class of $x$ to the $n$ ratios $x_j / x_i$ with $j \\ne i$ is a [[homeomorphism|0QT00PB5G2]], and the $n + 1$ sets $U_i$ cover $\\mathbb{RP}^n$.\n\n## Remarks\n\n$\\mathbb{RP}^1$ is homeomorphic to the circle $S^1$.\n$\\mathbb{RP}^2$ is the real projective plane, a closed surface that cannot be embedded in $\\R^3$.\n$\\mathbb{RP}^n$ is [[orientable|702SNFQG0W]] if and only if $n$ is odd.\nFor $n \\ge 2$ the quotient map $S^n \\to \\mathbb{RP}^n$ is the universal covering, so the fundamental group of $\\mathbb{RP}^n$ has order $2$.\n$\\mathbb{RP}^n$ is a CW complex with one cell in each dimension $0, 1, \\ldots, n$.\n\n## Explanations\n\n### L1 human\n\nTake a sphere and declare each point to be the same as the opposite point.\nThe space of these pairs of opposite points is the real projective space.\nEquivalently, each point of it is a line through the centre of the sphere.\nIn dimension one this gives a circle again: half of the circle, with its two end points glued.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "instance_of": [
      "QQQ9B1DK4P"
    ],
    "kind": "object",
    "msc": [
      "57N99",
      "51N15"
    ],
    "name": "Real projective space",
    "same_as": {
      "wikidata": "Q2597317",
      "wikipedia": "Real projective space"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Real_projective_space"
      }
    ],
    "tag": "1DDYB6VVSR"
  },
  "kind": "object",
  "link": "https://math2.tamnd.com/node/1DDYB6VVSR/",
  "links": [
    {
      "field": "instance_of",
      "target": "QQQ9B1DK4P"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/1DDYB6VVSR.md",
  "msc": [
    "57N99",
    "51N15"
  ],
  "name": "Real projective space",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/1DD/1DDYB6VVSR.md",
  "status": null,
  "tag": "1DDYB6VVSR",
  "uri": "m2://1DDYB6VVSR",
  "url": "https://math2.tamnd.com/wiki/real-projective-space/",
  "writing": {
    "tag": "1DDYB6VVSR",
    "kind": "object",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}