Why are manifolds defined with open sets?

Open sets (of fixed dimension) have a single "local model", namely an open ball. For closed sets, by contrast, matters are about as nasty as one can imagine:

  • In a Hausdorff space, points are closed; calling a Hausdorff space a "manifold" if for every point $p$, there exists a closed set $V$ containing $p$ and a homeomorphism from $V$ to some closed set in a Cartesian space is no condition at all. (Take $V = \{p\}$.)

  • Most closed sets are not the closure of an open set, e.g., non-empty closed sets with empty interior. The preceding point aside, there is no hope of deducing any kind of structure on a space locally modeled on closed sets.

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Differential Geometry

Soft Question

Surfaces

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