I think the term you wanted was "closed", which means does it curve enough (through higher dimensions) that it meets itself. It would only be closed if there is enough mass in the universe to warp space it on itself over the extent of the entire universe.
It will be a hot issue in about 60 Billion years when things might get really dicey in this reality.
If space is "open" then matter just keeps diffusing through the universe, particles decaying, until we're down to a loose collection of unbound fundamental particles and everything is slow, cold, and far away from anything else: universal heat death, or a "Big Freeze" - not a pleasant prospect.
If space has enough mass to be closed, everything collapses back on itself again in a maelstrom of energy and destruction: a "big crunch" or as Douglas Adams put it, a "gnab gib".
A relatively recent theory (which unfortunately seems to have some circumstantial observational evidence to back it up) is the "Big rip". In it, there not only may not be enough mass to close space, but that Dark Energy bends space the
other way (like
this) and is accelerating the expansion of space so much, that in 60 billion years everything (including fundamental particles themselves) are flying apart
so fast (faster than the speed of light) that they cannot have any interaction in any ways, not even to "see" each other, or to exert effects on each other. Each particle in our current universe becomes isolated in its own private "pocket" of space time. Even less fun than the "Big Freeze".
I'll just wait and see what happens in 60 billion years.
----
edited for typos