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Differentiable Atlas

A differentiable atlas of degree \(k\) of a manifold \((X,\mathcal{T})\) is a collection of ordered pairs ('charts') \(\mathcal{A}=(U_i,\phi_i)_{i\in I}\) where \(U_i\in\mathcal{T}\) and \(\phi_i:U_i\to\mathbb{R}^n\) is a homeomorphism onto its image \(\phi_i(U_i)\in\mathcal{T}_{\mathbb{R}^n}\) (where \(\mathcal{T}_{\mathbb{R}^n}\) is the metric topology on \(\mathbb{R}^n\)). The charts must also be \(C^k\) compatible, i.e. for every pair of charts \((U,\phi)\) and \((V,\psi)\), the transition map \(\psi\circ\phi^{-1}:\phi(U\cap V)\to\psi(U\cap V)\) must be \(C^k\) differentiable. The set of all degree-\(k\) atlases on \(X\) form a partial order, and a maximal element of this partial order (i.e. an atlas \(\mathcal{A}\) such that no other chart on \(X\) is compatible with every chart of \(\mathcal{A}\)) is called a \(C^k\)-differentiable structure on \(X\).