🚀 go-pugleaf

In article <SMITH.92Nov5101553@gramian.harvard.edu> smith@gramian.harvard.edu (Steven Smith) writes:
>pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>
>> What I don't understand here is how "living in R^n" is of itself
>> creating complexity and obscurity.  I can see that the
>> arbitrariness of f might get in the way.  But where does the
>> complexity and obscurity creep in if for example we define a
>> manifold to be a smooth retract of an open subset of R^n?
>
>The choice of ambient space is also somewhat arbitrary.  If the
>manifold can be imbedded in R^n, then it can also be imbedded in
>R^{n+1}, to say nothing of imbeddings in other manifolds.  The point
>is that manifolds exists independently of their imbeddings in R^n or
>other manifolds.  Any reasonable definition should possess this
>invariance.

This won't wash, you've just replaced one arbitrariness with another
more complicated one, namely the notion of atlas.

No one's mentioned atlases and charts in any of these posts,
understandably, so I'd better say what they are, and in so doing give
the full and I hope standard definition of abstract manifold (to within
a certain notion of smooth coordination I'll skip).  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.

Now why do you object to my arbitrariness and not the arbitrariness of
atlases?  You chop up a manifold in some arbitrary way into overlapping
pieces and then paste it back together with charts that coordinate
smoothly at the overlaps.  For example to chart a 2-sphere you might
chop it up into the northern and southern hemispheres, slightly
extended to open sets so that they overlap at the equator, and then
glue them back together "seamlessly".  But you might equally well chop
it into east and west, or into 240 pieces, one per country or sea.
What is nonarbitrary about this chopping?

The only arbitrariness in my definition is the choice of neighborhood
of the manifold and its retraction onto the manifold.  No chopping, no
charts.

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.
-- 
Vaughan Pratt				There's no truth in logic, son.