Manifold that is Hausdorff and second countable

Question

Why are we usually assume that a manifold $M$ has to be a Hausdorff space and Second countable ? Is it really hard to study smooth manifolds without making these assumptions?

Answer 1

Whitney's embedding theorem teaches us, that every manifold is locally the image of an $U\to \mathbb{R}^p$ smooth embedding (for a suitable p). That shows that the notion of manifold is quite natural. On the other hand Whitney's theorem doesn't hold without these assumptions.