Differentiable manifold
Patching together Euclidean pieces to form a manifold

One can reverse-engineer the above definitions to obtain one perspective on the construction of manifolds. The idea is to start with the images of the charts and the transition maps, and to construct the manifold purely from this data. As in the above discussion, we use the "smooth" context but everything works just as well in other settings.

Given an indexing set A, {\displaystyle A,} let V α {\displaystyle V_{\alpha }} be a collection of open subsets of R n {\displaystyle \mathbb {R} ^{n}} and for each α, β ∈ A {\displaystyle \alpha,\beta \in A} let V α β {\displaystyle V_{\alpha \beta }} be an open (possibly empty) subset of V β {\displaystyle V_{\beta }} and let ϕ α β: V α β → V β α {\displaystyle \phi _{\alpha \beta }:V_{\alpha \beta }\to V_{\beta \alpha }} be a smooth map. Suppose that ϕ α α {\displaystyle \phi _{\alpha \alpha }} is the identity map, that ϕ α β ∘ ϕ β α {\displaystyle \phi _{\alpha \beta }\circ \phi _{\beta \alpha }} is the identity map, and that ϕ α β ∘ ϕ β γ ∘ ϕ γ α {\displaystyle \phi _{\alpha \beta }\circ \phi _{\beta \gamma }\circ \phi _{\gamma \alpha }} is the identity map. Then define an equivalence relation on the disjoint union ⨆ α ∈ A V α {\textstyle \bigsqcup _{\alpha \in A}V_{\alpha }} by declaring p ∈ V α β {\displaystyle p\in V_{\alpha \beta }} to be equivalent to ϕ α β (p) ∈ V β α. {\displaystyle \phi _{\alpha \beta }(p)\in V_{\beta \alpha }.} With some technical work, one can show that the set of equivalence classes can naturally be given a topological structure, and that the charts used in doing so form a smooth atlas. For the patching together the analytic structures(subset), see analytic varieties.
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