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  "body": "\n## Definition\n\nLet $X$ be a topological space whose [[singular homology|M0Q5CFJ7ZH]] groups $H_k(X)$ are finitely generated for all $k$ and zero for all but finitely many $k$.\nThe *Euler characteristic* of $X$ is\n$$\\chi(X) = \\sum_{k \\ge 0} (-1)^k b_k(X),$$\nwhere $b_k(X)$ is the $k$-th [[Betti number|ST71ZV3FES]] of $X$.\n\n## Remarks\n\nIf $X$ is a finite [[CW complex|5W95QFHXPR]] with $c_k$ cells of dimension $k$, then $\\chi(X) = \\sum_k (-1)^k c_k$.\nFor the boundary of a convex polyhedron this is [[Euler's polyhedron formula|09HT3P35YP]] $V - E + F = 2$.\nFor any field $F$, $\\chi(X) = \\sum_k (-1)^k \\dim_F H_k(X; F)$ under the same hypothesis.\n\nExample: the torus has $b_0 = 1$, $b_1 = 2$, $b_2 = 1$, so $\\chi(T^2) = 1 - 2 + 1 = 0$.\nIts usual CW structure has one $0$-cell, two $1$-cells and one $2$-cell, and $1 - 2 + 1 = 0$ again.\n\nNon-example: the wedge of countably many circles has $H_1$ free abelian of infinite rank, so it fails the finiteness hypothesis and has no Euler characteristic in this sense.\n",
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