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Is there a generalization of the concept of manifold that captures the following idea:

Consider a sphere that instead of being made of a smooth material is actually made up of a mesh of thin wire. Now for certain beings living on the sphere the world appears flat and 2D, unware that they are actually living on a mesh, but for certain other smaller beings, the world appears to be 1D most of the time (because of the wire mesh).

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    Sounds like a job for topology.2011-12-14
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    Essentially, you'd want to define bounded subsets of the plane that have this property. It's hard to see how to do this purely topologically, without a metric, because you need a notion of scale, but you want the space to be, locally, a planar graph, with some density condition. The density condition seems to require a metric (and a specific embedding of a local planar graph onto a plane.)2011-12-14

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