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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. – Diego Dec 14 '11 at 18:11
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.) – Thomas Andrews Dec 14 '11 at 18:30

One thing to look at is foliations (and laminations), which are decompositions of manifolds into lower-dimension manifolds. While there is no "mesh" because each lower-dimension manifold has another lower-dimension manifold in any neighborhood, there is still a lower-dimensionality that is something like what you seek. (When you're looking at surfaces in a $3$-manifold, you can also look at the one-dimensional transversals.) See, e.g., H. B. Lawson, Foliations, Bulletin of the AMS 80:3 (1974), 369–418, MR 0343289 (49 #8031).

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I think that the concept that you want is that of a stratified space. Particularly, the stratification of a manifold by a piecewise linear (PL) decomposition.

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