Easy enough to understand, but more importantly, how is this
going to help to make the perfect cup of expresso?
Donovan Rebbechi <abuse@aol.com> wrote:
> On 2004-07-08, Virginiaz <virginiaz@aol.commentary> wrote:
> > Topology. Please discuss.
>
> What would you like to know about it ? Are you considering
> studying it ?
>
> Topologists are interested in studying topological
> invariants of various objects. That is, they look at
> properties that are independent of things like bending and
> stretching. If you can map one object to another with a
> 1-1 mapping that is continuous and has a continuous
> inverse, then to a topologist, they are the same thing (if
> two objects can be mapped such a way, they are
> homoemorphic, or one could simply say, topologically
> equivalent).
>
> So the topologist looks for algebraic and geometric
> properties that apply across a homoemorphism class. For
> example, Euler characteristic of a surface is topology
> invariant (so it's 2 for any polygon but 0 for a torus)
>
> Some of these may involve a certain amount of geometry.
> For example, knot theory looks at the space around the
> knot. The geometry of the embedding is important, even
> though the knot is always topologically a surface (though
> the way it is embedded varies)
>
> Another toplogical invariant is the number of connected
> components ("connected" means what you think it means. A
> connected component is a maximal connected subset)
>
> There are actually algebraic invariants of topological
> objects. For example, one could define a formal sum with
> integer coefficients assigned to each connected component,
> (x1 C1,x2 C2,x3 C3) where C1, ... , C3 are components and
> x1 are coefficients. Then one can define an operation +
> by: (x1,x2,x3) + (y1,y2,y3) = (x1+y1,x2+y2,x3+y3) This
> algebraic structure is a topological invariant (a
> "homology group" actually). Most of the work I did
> involved exploring algebraic invariants of topological
> objects.
>
> HTH,
