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From Wikipedia:

The broadest common definition of manifold is a topological space locally homeomorphic to a topological vector space over the reals.

A topological manifold is a topological space locally homeomorphic to a Euclidean space.

In both concepts, a topological space is homeomorphic to another topological space with richer structure than just topology. On the other hand, the homeomorphic mapping is only in the sense of topology without referring to the richer structure.

I was wondering what purpose it is to map from a set to another with richer structure, while the mapping preserves the less rich structure shared by both domain and codomain? How is the extra structure on the codomain going to be used? Is it to induce the extra structure from the codomain to the domain via the inverse of the mapping? How is the induction like for a manifold and for a topological manifold?

Thanks!

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    Yes, exactly: the extra structure on the codomain is pulled back to give additional local structure on the domain.2011-08-14
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    Thanks! How is the pull back like for a manifold and a topological manifold?2011-08-14
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    I find your question confusing. What else is there to do? If you're going to say some object $X$ is locally like object $Y$, how can you do that in a way that gives $X$ *more* structure than $Y$?2011-08-14
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    @Ryan: That is because I am confused. "If you're going to say some object X is locally like object Y, how can you do that in a way that gives X more structure than Y?" is also my question.2011-08-14
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    Well there's a simple answer to that. If $X$ has all the structure of $Y$, $X = Y$. So this is the most uninteresting case of a manifold.2011-08-14
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    @Dylan: What does "trivial " mean?2011-08-14
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    In some cases, the (local) similarity between the manifold and $\mathbb R^n$ does not extend beyondthe purely topological aspect, as in the graph of f(x)=|x| with the subspace topology; in this case, you can pullback the topology but not the differential part . In other cases, the local similarity does go beyond the topological and into the differentiable, so you can pullback both parts, like, e.g., on the n-sphere, and then you can do calculus on manifolds--by working in $\mathbb R^n$ and using charts to go back to the manifold.2011-08-14
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    Related: http://math.stackexchange.com/questions/53021/defining-a-manifold-without-reference-to-the-reals2011-08-14

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