🚀 go-pugleaf

In article <1992Nov3.204551.29715@lmpsbbs.comm.mot.com> bhv@areaplg2.corp.mot.com (Bronis Vidugiris) writes:
>Speaking of manifolds.....
>
>My mathematical education stopped well short of covering such topics.  Is
>the 'definition of a manifold' something that is feasible to 'pick up', or is
>it too involved?  If its possible to 'pick up', what would be a good, basic
>simple-as-possible source?

I wonder why you didn't post to sci.math.  I hope someone thinks of a
nice simple book -- it is not too hard to get the gist of manifolds, but
most books are more ambitious and throw in a lot of other stuff too.

I guess you may have the basic gut idea of a manifold, since you
referred to Lobachevsky.  Anyway, the idea is that a manifold is a
generalization of a surface to higher dimensions.  

Anyway, manifolds come in several flavors, most notably "smooth" and
"topological".  A topological manifold is a Hausdorff space such that every
point has a neighborhood homeomorphic to R^n (Euclidean n-space).  Now
you should ask me to define whatever didn't make any sense in that
definition!  Hopefully the point gets across, though: it's a space that,
in the vicinity of any point, looks like Euclidean space.  For example,
the surface of a doughnut looks, in the vicinity of each tiny ant on the
doughnut, like a vast plain, or plane -- R^2.  Similarly, space could be
not R^3 but something else that just looked like R^3 locally -- you'd
have to take an extensive rocket journey to see, for example, whether
space might not be an S^3, so that like Columbus one could head thisaway
and wind up coming around from thataway, having circumnavigated space.