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A manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n.

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Background to learn about manifolds

It's not really about manifolds (discusses them some) but maybe it's the kind of physics you're looking for. Full text is online and I always like it when they do that. …
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