On Wed, Aug 05, 2009 at 10:37:58AM -0600, Nicholas Thompson wrote: > There is a topologist on the list (at least one) who, I am hoping, will offer > at least one more definition of manifold. I say hoping, because at present, > I dont understand why "set" or "metaset" is not a perfectly good definition > of the non-roger definitions of manifold so far offered. >
My suggestion that the patches need to be continuous immediately rules out arbitrary (embedded) sets from being manifolds. As I said, I find it hard to conceive that a mathematician would call the Cantor set a manifold. It sounds like a tortuous abuse of language. It might even be that the patches need to be diffeomorphic, aside from a set of measure zero. This would allow the surface of a cube to be a manifold, but not say the boundary of the Mandelbrot set. Note that the only manifolds I ever studied were smooth manifolds (ie surface of a cube is not a smooth manifold). It seems Wikipedia only considers smooth manifolds too: http://en.wikipedia.org/wiki/Manifold But then the article http://en.wikipedia.org/wiki/Categories_of_manifolds explictly generalised the concept of smooth manifold (eg piecewise linear, topological, etc). It seems the concept is that for every point on the manifold, there is a neighbourhood N that is homeomorphic to to a Euclidean space R^n. Homeomorphic means the map f:N->R^n is continuous, but also its inverse f^{-1}:R^n->N. Cheers -- ---------------------------------------------------------------------------- Prof Russell Standish Phone 0425 253119 (mobile) Mathematics UNSW SYDNEY 2052 [email protected] Australia http://www.hpcoders.com.au ---------------------------------------------------------------------------- ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org
