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  "body": "\n## Summary\n\nSingular homology assigns to every topological space $X$ a sequence of abelian groups $H_n(X)$, built from all continuous maps of simplices into $X$.\nIt is defined for every space without any choice, it is functorial, and it is invariant under homotopy equivalence.\nIt satisfies the Eilenberg–Steenrod axioms, and on CW complexes and simplicial complexes it agrees with cellular and simplicial homology, which makes it computable.\n\n## Explanations\n\n### L1 human\n\nSingular homology looks at all the ways of placing points, paths, triangles and higher simplices into a space, however crumpled.\nIt forms formal sums of them and keeps those sums that have no boundary, modulo those that bound something bigger.\nWhat remains counts holes: $H_0$ counts path components, $H_1$ detects loops that do not bound, and so on.\nFor example, the circle has $H_0 \\cong \\Z$, $H_1 \\cong \\Z$ and all higher groups zero.\n",
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