Learn more physics!
Assume there's no expansion of the universe. Is it true to say that in a closed universe, a particle travelling always in the same direction will end up at its starting point? And a particle travelling in the same direction in an opened universe will end up getting farther from its starting point quickier than it moves away from it without exceeding the speed of light i.e. the distance from the reference point exponentially increases while the particle's speed and time reference stay constant?
You've got the general picture, but I would rephrase one thing. Space doesn't have a coordinate system all laid out in it. So it doesn't really mean anything to say that a particle ends up at its "starting point". What you could do is take two particles starting together but in non-accelerated uniform motion. They will end up rejoining in a closed universe.
As for two particles moving away in a simple expanding universe with no cosmological constant in a conventional coordinate system their relative velocity will slightly shrink, because gravity is slowing the expansion. At some point in a universe like ours with accelerating expansion the particles will be outside each others horizons. That sort of conclusion, like the conclusion that two particles in the closed universe re-collide, is independent of choice of coordinate systems.
(published on 08/11/10)
Follow-up on this answer.