Posted by
Russell Standish on
URL: http://friam.383.s1.nabble.com/manifold-in-mathematics-tp3385914p3395392.html
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/ManifoldBut 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
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