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Is the conclusion that a manifold is a space that looks like a lower dimensional space that it really is?

Yes and no. The definition doesn't require that what you have is embedded in a higher dimensional space. Though that's a good source of examples.


Basically says that many manifolds can be described as a lower dimensional submanifold of some euclidean space.

no. there's no requirement for lower dimension. like another person that responded to you R^3 is 3 dimensional manifold. a (smooth) manifold is a space you can do calculus on. that means you have a consistent notion of distance (so that you can figure out when points are close together, ie you approach limits). the set of charts and continuous transformations between them is what encodes this constraint.

You can have a thing in some higher dimensional space that still "looks like" a thing in lower dimensional space (like the surface of the sphere in R^3 being two dimensional), but R^3 itself is a 3 dimensional manifold that "looks" 3 dimensional as well.

To some extent, yes! It needs to have a 'nice' structure, though---this is why we have the condition that is should resemble, say, an Euclidean space.

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