🚀 go-pugleaf

In article <1992Nov5.060400.14203@CSD-NewsHost.Stanford.EDU> pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:

> Ah, now this is starting to sound very interesting and helpful.  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 problem with that is that it focuses too much on an aspect of a
manifold that just isn't terribly fundamental.

For example: the 2-sphere S^2 and a Klein bottle are both perfectly
nice two-dimensional manifolds.  However, S^2 can be embedded in R^3,
and a Klein bottle has to be embedded in R^4.  There's nothing
particularly fundamental or interesting about that difference, and it
would be a shame to have a definition of a manifold that obscured the
essential similarity between these two two-dimensional manifolds.
--
Matthew Austern                   Just keep yelling until you attract a
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