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  "body": "\n## Summary\n\nThe $k$-th Betti number $b_k(X)$ of a topological space $X$ is the rank of its $k$-th integral [[singular homology|M0Q5CFJ7ZH]] group.\nInformally, $b_0$ counts the path components, $b_1$ counts independent loops that do not bound, and $b_2$ counts enclosed cavities.\nBetti numbers are homotopy invariants, and their alternating sum is the [[Euler characteristic|RNDDWDAFJ5]].\n\n## Explanations\n\n### L1 human\n\nBetti numbers count holes of each dimension.\nA circle has one piece and one loop around a hole, so $b_0 = 1$ and $b_1 = 1$.\nThe surface of a doughnut has one piece, two independent loops (around the tube and around the central hole) and one enclosed cavity, so its Betti numbers are $1, 2, 1$.\nA solid ball has $b_0 = 1$ and no holes at all, so its other Betti numbers are zero.\n",
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