🚀 go-pugleaf

In article <SMITH.92Nov5132141@gramian.harvard.edu> smith@gramian.harvard.edu (Steven Smith) writes:
>For argument's sake, lets call a locally Euclidean space a manifold,
>and a locally Euclidean space endowed with a differentiable structure
>(atlas) a differentiable manifold.

Just for the record, a manifold is a HAUSDORFF locally Euclidean space.
You want to exclude examples like the real number line with two origins.
(X= (Rx{0,1})/~ where (a,b)~(c,d) iff a=c and either b=d or a=c <>  0.)

As for the arbitrariness of the atlas, I have seen manifolds defined as
a pair (M,F) where F is a _maximal_ family of compatible charts. Then
in any application one selects a collection of charts which covers M
and is handy for the application, without making the choice of chart
be an explicit part of the manifold's construction. I don't find this
to be a very natural perspective, but if you like Bourbaki...

dave rusin@math.niu.edu