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  "body": "\n## Summary\n\nA topological manifold of dimension $n$ is a [[topological space|0CQHFPGJ45]] that looks like the Euclidean space $\\R^n$ near each of its points.\nIt is also required to be Hausdorff and second-countable.\nIt is the basic object of geometric topology.\nCurves, surfaces and the space-times of physics are manifolds, and smooth and piecewise linear manifolds are topological manifolds with extra structure.\n\n## Explanations\n\n### L1 human\n\nImagine a tiny ant on a surface: around it, the world looks like a flat plane, even though the whole surface may be a sphere or a doughnut.\nA manifold is a space with this property in some fixed dimension $n$.\nThe circle is a $1$-dimensional manifold, because a short arc of a circle looks like a short open interval.\nThe surface of the Earth is a $2$-dimensional manifold, which is why flat maps of small regions work.\nA figure eight is not a manifold, because near the crossing point four arcs meet, which no open interval can imitate.\n",
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