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It seems like the "locally Euclidean" condition in the definition of a manifold is most important and most frequently used. I've also seen the standard examples of spaces which are locally Euclidean, but not second countable or Hausdorff. I'm wondering how much of differential geometry can be recovered if we only assume the locally Euclidean condition. For example, could we still prove a version of Stokes theorem, etc.

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I think it is used with other properties to show paracompactness, which is used to show the existence if partitions of unity.POUnity help glue together objects like vector fields that are defined only locally, i.e., in individual charts. An example is the proof that every $C^\infty$ manifold admits a Riemannian metric: you pull back the inner-products (defined on the chart image in $\mathbb R^n$) using chart maps, which gives you a collection of inner-products defined on individual charts. The POU help glue this collection of local objects into a single , global object defined on the manifold.

To see how 2nd countability is used to show the existence of POU's , See, e.g,: 4.9 thru 4.12 in:

https://www.math.lsu.edu/~lawson/Chapter4.pdf

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