Re: Examples
F. Javier Thayer (jt@linus.mitre.org)
Fri, 17 Jun 1994 18:01:07 -0400
I believe I implied that the theorem in Trybulec's example is highly
non-trivial. I do claim however that one can do rigorous mathematics
that would appeal to a wider community of individuals with less work.
I don't believe we should take the alternate approach of doing
partititions of unity on differentiable manifolds either. I mentioned
this as an explanation of the fact that most mathematicians are
terribly interested in this result at all and get can get by without
ever having heard of it.
As far as the 'little theories' required to prove unity
partitioning for differentiable manifolds, I don't know the answer,
but it obviously requires a lot more stuff than the metric space
approach Trybulec mentioned and is not a good candidate for a test
case. Notice, however, it falls almost for free from other
developments in a standard differentiable manifolds course (actually
the fact that manifolds are locally compact helps), and one also gets
better properties for the partitions.
Javier