questions:
----------
1) does the 
cosmological principle introduce any restrictions on the topology of the 
universe.
2) does the topology science relies on some multidimentional theory 
- like string theory
 
now look here,
 
the problem:
------------
 
I finally figured it out. It is about that 
transparency problem. The problem was how it is possible to roll up the 
transparency into a torus not stretching it so  it's curvature remained the 
same. I was stubborn to say that this was not possible, so we came to the 
conclusion that the 2-torus in three dimenitonal eucliean space is not an good 
example of two dimentional closed space of zero curvature. But if we introduced 
the fourth spital dimention than it becomes possible. Like for example we 
imagine a string which has one pair of zero dimentional ends, we need at least 
one more dimentional space (it's two) to make it closed (to joing the ends), so 
in  case of a flat plane (or paper) which has two pairs of one dimentional 
ends which we have to join, then we need two more dimentions to do that. That 
would be 4 dimentions. Thus in four dimenitons matematically we can do with the 
topology of the space whatever we want and it's curvature will still be the 
same. 
 
But the problem now is how to put this idea 
into the real world. All these considerations about the topology of space were 
made from "the higer groud" - I mean from the point of view of a spce with at 
lest two dimentions more that the space (earlier a transparency or just a piece 
of paper) we were thinking of. The problem appeares if we talk about the 
topology of the Universe where there are only three spital dimentions. To make 
it close like 3-torus it is essential to introduce 3 more dimentions (it's 
six) to make it possible. Then I say that, that all topoloy thing implies that 
out phisical space is in fact more than three dimentional, otherwise we would 
stand before question: what is that thing that out space (universe) lies in. 
Since this question is considered to be sensless, or at least redundant, the 
only solution is to assume the first option.
It leads to the link with the 
science of the nature of the space itself - like in the string theory, which 
assumes that our spacetime is 10-dimentional - time plus three known spital 
dimentions and 6 other which were reduced to the lenght of Planck during the 
very early stages of evolution of the universe. In that case that bending our 
unverse into a 3-torus around those additional axies could be easily (al least 
theoreticaly) done but the "radius" of that bend would be very small. But 
whatever.
Is this way of thinking of topology ok 
?
further 
questions:
------------------
 
For the curvature matter, there is general theory of relativity which 
connects the energy density with curvature. 
What about the topology ? What is that force that shapes the topology of 
space, and can the topology change in time ?
 
pozdrawiam 
Bartek