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  "body": "\n## Summary\n\nThe Euler characteristic $\\chi(X)$ of a space is the alternating sum $b_0 - b_1 + b_2 - \\cdots$ of its [[Betti numbers|ST71ZV3FES]].\nFor a finite [[CW complex|5W95QFHXPR]] it equals the alternating count of cells, which recovers $V - E + F$ for polyhedra.\nIt is a homotopy invariant, and it is multiplicative for products and finite coverings.\nIt links topology with geometry through results such as the Gauss–Bonnet theorem.\n\n## Explanations\n\n### L1 human\n\nCount the vertices, edges and faces of a convex polyhedron, and compute $V - E + F$.\nThe answer is always $2$, whatever the polyhedron, for example $8 - 12 + 6 = 2$ for a cube.\nThis number belongs to the sphere, not to the particular polyhedron.\nFor the surface of a doughnut the same count always gives $0$.\nThe Euler characteristic extends this count to any reasonable space, using homology in place of counting cells.\n",
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