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By locally Euclidean space I mean a topological space which is locally homeomorphic to $\mathbb{R}^n$. Smooth locally Euclidean space is a locally Euclidean space along with a smooth atlas. Smooth manifolds can then be characterized as second countable Hausdorff smooth locally euclidean spaces. However, how necessary are the extra topological restrictions to develop differential geometry. In particular, local constructions such as tangent vectors should hold without them. For example, it seems to me that a book such as An Introduction to Manifolds by Tu maybe developed without the topological restrictions up to chapter 6 on integration. Is this correct?

Ivan Burbano
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  • Possible duplicate? https://math.stackexchange.com/questions/2131530/why-is-important-for-a-manifold-to-have-countable-basis – preferred_anon Dec 24 '19 at 15:42
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    A smooth locally Euclidean space has a partition of unity subordinate to every open cover iff it is Hausdorff and second countable, so any argument using them needs to be checked – Alessandro Codenotti Dec 24 '19 at 15:45

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