🚀 go-pugleaf

Vaughn Pratt asks why the modern abstract definition of a manifold is
"better" than an "imbedded" definition.

The theory of Riemann surfaces surely provided important motivation,
historically, for the modern approach.  For example, consider the classic
situation for elliptic functions: you can start by constructing the branched
two-cover of the Riemann sphere associated with the equation 
w^2 = (z^2-1)(k^2 z^2 -1), and take integrals on that, but you only really
see what's going on when you recognize that this surface is conformally
equivalent to a torus.

Now of course you could pick a particular imbedding of the torus in 3
space, and a particular immersion of the branched surface in 3 space, but
this emphasizes the incidental and obscures the essential.

If you'll buy this argument for conformal atlases, why not for smooth ones?
(It would be interesting to know which came first.)

I believe there is a historical connection between this line of thought and
the origin of the modern definition of a topological space, but I better
not say any more without checking.