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## Main Question or Discussion Point

In general terms a manifold can be defined simply as a topological space locally resembling Euclidean space with the resemblance meaning homeomorphic to Euclidean space, plus a couple of point set axioms that avoid certain "patological" manifolds and that some authors reserve for the definition of differentiable manifolds. My doubt is that since the category of homeomorphisms doesn't include notions of distance and angles, that is, metric properties are not included, wouldn't it be more precise defining manifolds as topological spaces locally homeomorphic to real topological vector spaces (or complex topological vector spaces in the case of a complex manifold) rather than to Euclidean spaces?

Not that the usual definition is wrong, but IMO it also might be misleading wrt the often ignored difference between real topological vector spaces and Euclidean vector spaces (only the latter has the Euclidean inner product).Calling both entities R^n doesn't help either.

In the wikipedia page on manifolds one can see it defined both ways but there is much more insistence in the "locally resembling Euclidean space" definition.

Not that the usual definition is wrong, but IMO it also might be misleading wrt the often ignored difference between real topological vector spaces and Euclidean vector spaces (only the latter has the Euclidean inner product).Calling both entities R^n doesn't help either.

In the wikipedia page on manifolds one can see it defined both ways but there is much more insistence in the "locally resembling Euclidean space" definition.

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