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In defining a coordinate chart,

[tex]\left ( U,\phi \right ), U \in M, \phi : U \to \mathbb{R}^{n},[/tex]

on a manifold [tex]M[/tex], what exactly is [tex]\mathbb{R}^{n}[/tex]: the set of all n-tuples, a topological space, a metric space, a vector space, Euclidean space conceived of as an inner product space, Euclidean space conceived of as an affine space...? Or if [tex]\mathbb{R}^{n}[/tex] can mean different things for different manifolds, what is the minimum structure required of [tex]\mathbb{R}^{n}[/tex] for [tex]M[/tex] to be a manifold when defined by an atlas of such charts?

[tex]\left ( U,\phi \right ), U \in M, \phi : U \to \mathbb{R}^{n},[/tex]

on a manifold [tex]M[/tex], what exactly is [tex]\mathbb{R}^{n}[/tex]: the set of all n-tuples, a topological space, a metric space, a vector space, Euclidean space conceived of as an inner product space, Euclidean space conceived of as an affine space...? Or if [tex]\mathbb{R}^{n}[/tex] can mean different things for different manifolds, what is the minimum structure required of [tex]\mathbb{R}^{n}[/tex] for [tex]M[/tex] to be a manifold when defined by an atlas of such charts?

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