Posted: Dec 26, 2010 3:20 pm
by The Damned
twistor59 wrote:
The Damned wrote:Ok I'm a bit numpty. Can someone break this down in simple calculus terminology because I'm having a hard time following what exactly is meant by the topology.

Defining the topology of a set is basically giving it the property of a shape. Well almost - when you've defined its topology, you've only defined the shape up to equivalence under continuous deformations. By this I mean that two sets A and B "have the same topology" if you can continously deform A into B. Continously means without ripping holes in it, or gluing bits together.

Take for example this set:

{x ∊ ℝ s.t. (0<=x<=1) }

i.e. just the real interval between 0 and 1. You can change its topology by gluing ("identifying") the points zero and one. The set then has the topology of a circle. If you head on up towards 1, you eventually pop out at 0 again.

Topologies are rigorously defined using the concept of open sets. (Tons of stuff on it via google).

Sets in simple spaces like ℝN come ready equipped with an "obvious" topology - the metric topology, where the open sets are defined in terms of the distance measures obtained from the metric.

The Damned wrote:
I understand eigen states you don't have to dumb it down too much but those eigenvectors look strange!?

I mean they appear not to be the usual uniform vectors but seem to be arbitrarily non Euclidean in form. Why is that and am I just wrong?

Vector mathematics is ok but I haven't studied tensors yet so if you want to expand my matrix understanding that's fine. just to set some parameters.

Which eigenvectors look strange ? The eigenvectors in the LQG posts are elements of Hilbert spaces and you can add, subtract and multiply by scalars just like any ordinary vector space. The fundamental difference is that the Hilbert spaces may be infinite dimensional.

Have you ever considered teaching physics at university that was very straight forward. So straight forward that Google is unnecessary. Thanks that was worth waiting for. :D

So basically if we define a deformation axiomaticaly set theory can then be made compatible in any framework euclidean or not or did I get that wrong?

Would it be useful to say that infinitely eliptical or not ie there is no limit to the amount of bending we can do as long as we specify limtis? Maths is useful here in that respect?