🚀 go-pugleaf

In article <1992Nov6.190913.18507@nas.nasa.gov> asimov@wk223.nas.nasa.gov (Daniel A. Asimov) writes:
>HERE is an example to illustrate the pitfalls of the "extrinsic" approach:
>
>	Consider finite matric spaces.  One could use the approach of defining them
>	as any finite subset of Euclidean space with the inherited distance function.
>	Or one could use the abstract approach of a distance function with the usual
>	metric space axioms.
>
>	However, the first approach will fail to include some (usual) metric spaces 
>	containing only 4 points:  Let X = {A,B,C,D} with d: XxX -> R the metric 
>	defined via d(A,B) = d(B,C) = d(C,A) = 7 and d(A,D) = d(B,D) = d(C,D) = 4.
>	It is not difficult to show that this metric space cannot occur as a subset 
>	of a Euclidean space of any dimension.  (Example due to Richard Schroeppel)

This is very nice!  I wish there were as beautiful an example for
manifolds.  The thing about metric spaces is that the abstract
definition is *very* beautiful (I like generalizing this by dropping
symmetry and "d(x,y)=0->x=y" since this is yet simpler and moreover
then admits many objects that arise in computer science).  If there
were as simple and beautiful an abstract definition of manifold as
there is of (generalized) metric space you wouldn't hear any complaints
from me.  You can't make analogies like these stick in situations like
these!

(Beautiful: I said it thrice, beauty must be truth.)
-- 
Vaughan Pratt				There's no truth in logic, son.