Why do we generally require that a differentiable manifold be Hausdorff and second countable? Is this universally accepted in the definition? My Professor only required the Hausdorff condition for example, but most book I have read require both. Thanks
Those conditions provide the metrizability of the manifold. That is we want a metric space as a result. Actually we want to produce a Riemannian manifold. Some texts require paracompactness, which provides partitions of unity and helps to define a Riemannian metric. Look at Urysohn metrizatiın theorem and other metrization theorems, you will see that these conditions are in their hypothesis.