Re: what is the meaning (exact defintion) of 'times come to an end'?

Sorry, but I could not find this quote. I would guess it is areference to 
the fact that space-time could be curved in on itself (if the universe is 
denser than the critical density), so that the universe eventually comes 
together in a 'big crunch' which would end time.
Here is my answer to another question on general relativity, which gives a 
(very) brief outline of the subject.

Space-time can be curved. It is curved by mass (or energy) - this being 
the source of gravity. To warp it very much, a huge amount of mass would 
be required - this is what happens in a Black Hole.

If you drew a circle on a flat piece of paper, and measured the radius and 
the circumference, you would hope to find that 
	radius = circumference / 2*pi.
What if you were not aware of height, but lived on the Earth. You could 
draw the equator - a circle. The radius should be measured from the centre 
of the Earth. But being unaware of the direction 'down', you would 
measure the radius from the North Pole. This would be much larger than the 
actual radius. When you divided the equator (your circumference) by 2*pi, 
you would find that your measured radius was much bigger than your 
expected radius. There is an excess radius because the Earth's surface - 
your space - is bent.

The surface area of a sphere is A=4*pi*(r^2). If you measured the surface 
Area, you would expect to predict the radius, r=sqrt(A/(4pi)).
...unless, that is, our space-time is warped. In this case, there may be 
an excess radius.
Excess Radius = Predicted Radius - Measured Radius
	      = GM/(3c^2),
where G is Newton's Gravitational Constant, c is the speed of light, and M 
is the mass inside the sphere (assumed to be evenly distributed).
G/3c^2 has a value of about 2.5*10^-29 cetimetres per gram, so for each 
gram, there is an excess radius of about 2.5*10^-29 cm.
As you can see, a lot of energy is needed to warp space-time by a large 
amount.

I suggest you read 'Hyperspace' by Michio Kaku for a good account of this 
area, or 'A Brief History of Time' by Hawking, or perhaps 'Six Not-So-Easy 
Pieces' by Feynman, whose explanation of General Relativity I have just 
given you.

[Moderator's note:  Dan gives a great explanation of the basics of curved 
space.  When we apply general relativity to the whole universe, it turns out 
that it's possible that space can be like a sphere - this is called a "closed 
universe".  In this case, it turns out that such a universe will collapse in a 
"Big Crunch," exactly the opposite of a "Big Bang."  This would mean that time, 
at least as we know it, would end.  However, recent observations suggest that 
the universe will expand forever and that time will never end.  You can find 
more information by searching our site under "Big Crunch."]