🚀 go-pugleaf

In article <SMITH.92Nov5132141@gramian.harvard.edu> smith@gramian.harvard.edu
(Steven Smith) writes:

>No one ever said that arbitrariness is necessarily bad;

Actually, a couple of us have implied it.

>Sorry, but what is your definition specifically?  If manifolds are
>simply retracts of open sets of Euclidean space, then using this
>definition, how is projective space a manifold ?

This is not a problem, as long as you embed in a high enough space.  The
projective plane, for instance, embeds nicely in R^4 and it's easy to make
it into a retract of some tubular neighborhood.

I don't think that people are giving enough credit to the retract
definition.  It is often very convenient to be able to introduce a tubular
neighborhood around a manifold.  The Jordan curve theorem is one example.
With the classical approach, proving that manifolds can be embedded in
Euclidean space is quite a major theorem.  You do get some things almost
for free with the retract definition.  But as has already been pointed
out there are other limitations to this approach.
-- 
Tim Chow     tycchow@math.mit.edu
Where a calculator on the ENIAC is equipped with 18,000 vacuum tubes and weighs
30 tons, computers in the future may have only 1,000 vacuum tubes and weigh
only 1 1/2 tons.                               ---Popular Mechanics, March 1949