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?
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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.
Leonhardt von M
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So these assumptions rule out manifold that that isn't "look like" $\mathbb{R^n}$ in some sense? – SamC Mar 29 '15 at 13:24
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These assumptions rule out manifolds which are too abstract in the mean that they cannot be embedded in a finite dimensional euclidean space. You can imagine the manifolds as sets in form {f(x)=0} in $\mathbb{R}^p$ where the rank of the Jacobian of f is constant (and less than p). (the Implicit Function Theorem gives us an atlas) – Leonhardt von M Mar 29 '15 at 14:12