The definition of having multiple differentiable structures is that given two atlases, [tex]{(U_i ,\phi_i)} [/tex] and [tex]{(V_j,\psi_j)} [/tex] (where the open sets are the first entry and the homeomorphisms to an open subset of R(adsbygoogle = window.adsbygoogle || []).push({}); ^{n}are the second entry), that the union [tex]{(U_i,V_j;\phi_i,\psi_j)} [/tex] is not necessarily an atlas. Those atlases whose union are atlases form an equivalence class and define one type of differentiable structure.

But how can a manifold depend on the atlas? For example, how can two different explorers navigate the same world and draw up two inequivalent atlases?

Also, what is the relationship between having multiple differentiable structures and diffeomorphisms? The book I'm reading seems to imply that all homeomorphisms are diffeomorphisms if there is only a single differentiable structure. Is this true, and why? It seems to me there should be no relation between homeomorphisms and diffeomorphisms, since homeomorphisms only ask for continuity (of the map and inverse map between manifolds), but diffeomorphisms require differentiability.

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# Differentiable structures and diffeomorphisms

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