If not, does the second law of thermodynamics even apply? What role would entropy play if it is not?
To our knowledge it is. At minimum you could count the observable universe as a closed system because anything outside it will not have had time to affect you locally due to the finite speed of light.
We once assumed that we would fall off the edge of the earth too. When you say "observable universe" you refer to technology constraints. If all assumptions are based on limits in technological capability, what is the point in exploration and creative thought?
No, I refer to the actual observable universe in visible light or neutrinos (The latter is not actually possible at the moment). Before about 300,000 years after the universe started, it was too hot and too dense for light to move freely throughout the universe. Once it cooled off enough for nuclei to permanently combine with free electrons the universe became transparent to EM radiation, aka light, and the CMB (cosmic microwave background) was created at this time. So the furthest we can look back using light is to about 300,000 years after the creation of the universe as we know it. (I purposely don't use the term "big bang" because it inherently creates incorrect views on what happened) Prior to 300,000 years neutrinos were able to move freely since they interact very weakly with other matter, but we cannot observe them very well currently. Still, this puts a limit on how far back in time, or in distance we can see in either case.
The distance at which the original space that emitted the current CMB is believed to be about 14 billion parsecs, or 45.7 billion light years. The edge of the actual observable universe, with ANY form of radiation, is believed to be about 14.3 billion parsecs, 46.6 billion light years. (So the diameter of the observable universe is currently about 93.2 billion light years) Past this point we cannot see, even in principle, as nothing has had time to reach us yet.
Isn't the universe closed system by definition?
In this context the discussion alludes to a multiverse. In that case AFAIK the word universe reduces from it's definition of everything.
Ok so, is the universe a perpetual motion machine? See where I'm going?
If the universe is a closed system, where are the boundries?
Why would it be? Even though the total amount of energy remains the same the total amount of energy available for work always decreases.
It doesn't have any. Likewise it has no centre.
In order for it to be a closed system, it would require boundries.
That only applies if there is both a surrounding and a system but the universe has no surrounding. If it is infinite then the reason is obvious and even if it was finite it would wrap around itself (like how in the old fashioned video game asteroids if the player drove off through the right wall they would come out of the left).
Nope. The solution to the Einstein field equations for an expanding space-time is the FLRW metric. This space-time is spatially symmetric.
Another consequence of the FLRW space-time is that, as Ryan points out, it has no boundaries. It could be infinite. If it is finite, it's either a simply connected surface or a non-simply connected surface.
As Darkkith explained, because of the rate at which the universe expands, we can approximate our observable universe to be a closed system. So yes, it obeys the second law.
What about virtual particles? Wouldn't the existence of virtual particles imply that the universe is not closed? Virtual particles seem to come from outside of the universe and appear inside it.
Virtual particles exist only to mediate interactions. They disappear before they can do anything else, so no energy conservation laws are violated.
I think he's on to something. The universe by definition contains all there is. So if we were to witness something that appeared to emerge from nowhere, we would just say that it emerged from some other unknown place that is part of our universe. We would say that it popped out of a wormhole whose other side was in some other part of the universe. We would employ the same tactic Wolfgang Pauli used when he explained what happened to the missing energy in certain particle collisions, ie, it was carried away by a neutrino, though he used the word neutron. We wouldn't give up the principle of the conservation of matter we would invent some desperate loophole to save the theory.
In a way, that the universe is a closed system can become a useless tautology. Let's rename universe "everything" and let's replace the predicate: is a closed system with contains everything. After all, if something contains everything then beyond its borders is nothing. (though you could make the argument that speaking of the universe's borders is not sensible but you know what I mean) So now if you put the new subject and predicate together you get a useless tautology: everything contains everything.
Obviously if we witnessed a violation of one of the conservation laws and saw something appear from nothing we would have to change our science. The rest of your post makes no sense.
Separate names with a comma.