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  "body": "\n## Definition\n\nLet $X$ be a topological space, let $k \\ge 0$ be an integer, and let $H_k(X)$ be the $k$-th [[singular homology|M0Q5CFJ7ZH]] group of $X$ with integer coefficients.\nThe *$k$-th Betti number* of $X$ is the [[rank|EMCTDD3HZ1]] of the abelian group $H_k(X)$:\n$$b_k(X) = \\rank H_k(X) = \\dim_{\\Q} \\left( H_k(X) \\otimes_{\\Z} \\Q \\right),$$\na nonnegative integer or $\\infty$.\n\n## Remarks\n\nWhen $H_k(X)$ is finitely generated, it is isomorphic to $\\Z^{b_k} \\oplus T$ with $T$ finite, so $b_k$ is the number of copies of $\\Z$.\nBy the universal coefficient theorem, $b_k(X) = \\dim_{\\Q} H_k(X; \\Q)$.\nTorsion is invisible to Betti numbers.\n\nExample: for the torus $T^2 = S^1 \\times S^1$, $H_0 = \\Z$, $H_1 = \\Z^2$ and $H_2 = \\Z$, so $b_0 = 1$, $b_1 = 2$, $b_2 = 1$.\n\nNon-example of a finite Betti number: an infinite discrete space $X$ has $H_0(X)$ free abelian of infinite rank, so $b_0(X) = \\infty$.\nFor the real projective plane, $H_1 = \\Z/2$ is nonzero but has rank $0$, so $b_1 = 0$.\n",
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