🚀 go-pugleaf

In article <1992Nov5.161930.21320@CSD-NewsHost.Stanford.EDU>, pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
> 
> In article <SMITH.92Nov5101553@gramian.harvard.edu> smith@gramian.harvard.edu (Steven Smith) writes:
> >pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
> >
> This won't wash, you've just replaced one arbitrariness with another
> more complicated one, namely the notion of atlas.

An individual atlas may be arbitrary, but the manifold is not. A differentiable
manifold is defined as a set equipped with a COMPLETE atlas. That is
it contains every compatible chart.

One atlas is rather like a base for a topology - it describes the manifold
completely because there is only one completion just as a base for a
topology generates a unique topology.

> An ATLAS F on a
> topological space M is a family of CHARTS (fi,Ui) where the family <Ui>
> is an open cover of M and each fi:Ui->R^n is a homeomorphism from Ui to
> an open subset of R^n, such that the fi's of overlapping Ui's
> "coordinate smoothly" at the overlap.  A MANIFOLD is a pair (M,F)
> consisting of a topological space M and an atlas F on M.

You don't need to build homeomorphisms into the definition. The fi
define a topology such that they become homeomorphisms. There is
no real practical impact to this - either you define the topology first
and then define the manifold structure, or you define them simultaneously.
 
> 
> Furthermore my definition is a *lot* simpler, I'd say, even without
> going into the notion of smooth coordination required to define an
> atlas.
> 
> There has to be some other reason why the complicated notion of atlas
> is essential.

In what way is it complicated? A chart seems to be a very natural
expression of the idea of locally Euclidean.

Terry Moore