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  "body": "\n## Statement\n\nLet $S$ be a closed [[surface|VGJJVBZ1MQ]], that is, a compact connected surface without boundary.\n\n1. $S$ is homeomorphic to exactly one surface in the following list.\n   - The sphere $S^2$.\n   - The [[connected sum|Y0K0K5NGWW]] of $g \\ge 1$ copies of the [[torus|03NM7DZXXC]] $T^2$.\n   - The connected sum of $k \\ge 1$ copies of the real projective plane $\\R P^2$.\n2. Two closed surfaces are homeomorphic if and only if both are [[orientable|0TD904MG06]] or both are non-orientable, and they have the same [[Euler characteristic|RNDDWDAFJ5]].\n\nThe Euler characteristic of the connected sum of $g$ tori is $2 - 2g$, and that of the connected sum of $k$ projective planes is $2 - k$.\n\n## Remarks\n\nIdea of the proof: by Radó's theorem every surface has a triangulation, so $S$ is obtained from a polygon by gluing its edges in pairs.\nCutting and regluing the polygon brings the gluing word to a normal form: $aa^{-1}$ for the sphere, $a_1 b_1 a_1^{-1} b_1^{-1} \\cdots a_g b_g a_g^{-1} b_g^{-1}$ for $g$ tori, or $c_1 c_1 \\cdots c_k c_k$ for $k$ projective planes.\nThe surfaces in the list are different because orientability and Euler characteristic are topological invariants that take different values on them.\n\nThe theorem at work: the [[Klein bottle|6J4C2P3ZHC]] is a closed non-orientable surface with Euler characteristic $0$, so it is homeomorphic to $\\R P^2 \\# \\R P^2$.\nAlso $T^2 \\# \\R P^2$ is non-orientable with Euler characteristic $-1$, so it is homeomorphic to $\\R P^2 \\# \\R P^2 \\# \\R P^2$.\nCompactness is needed: the open disc and the open annulus are connected surfaces without boundary that are not in the list.\n\n## Explanations\n\n### L1 human\n\nEvery closed surface is a sphere with some number of handles, or a sphere with some number of crosscaps.\nTo tell two closed surfaces apart, two pieces of information suffice: whether they have a consistent inside and outside, and their Euler characteristic.\nSo the infinite variety of possible shapes reduces to a simple list.\n\n### L2 human\n\nThe classification is the model of what one hopes for in manifold topology: a complete list with computable invariants.\nThe orientable number $g$ is the [[genus|NJ787GZS1N]] of the surface.\nThe proof through polygons and gluing words also computes the fundamental group of each surface.\nEvery closed surface other than $S^2$ and $\\R P^2$ is aspherical, and every closed surface carries a metric of constant curvature, with sign given by the Euler characteristic.\nIn dimension $3$ no list this simple exists, and the analogue is the [[geometrization conjecture|B27HB43K86]].\n",
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