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I am currently reading 'Geometrical Methods of Mathematical Physics' by Bernard Schutz and I have some questions about manifolds. I'm fairly new to Differential Geometry so bear with me!

On P33 he states that 'manifolds need have no distance relation between points, we shall need a definition of a vector which relies only on infinitesimal neighborhoods of points of M'.

My question is: how can you define the neighborhood around a point if you haven't already defined a distance relation between points? When you define a manifold are you simply defining a set of points which have a 1-1 mapping to Euclidean space or are you also defining a distance relation between the points?

Thanks in advance,

Pete

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# Do Manifolds have distance relations between points?

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