In the very near future a NASA probe returns to Earth with a soil sample from an asteroid containing a strange substance able to catalyze cold fusion reactions. The economic value of this substance makes space exploration for more of this element promising.Soon, returns are soo high, that mining becomes a profitable business, resulting is dozens of Governments and Corporations pouring hundreds of billions of dollars into an exploration fund to search for more of the element.
Within a few years, the human population living in space explodes into the tens of thousands. While many of the colonies are mostly self-sufficient they were made with the expectation that they could be resupplied every few years. That said, the technology that went into them is not significantly more advanced than what is available today. The only major things that sets them apart from modern space stations is that they have the ability to mine and refine ore from asteroids down into their constituent parts, and the fusion catalyst gives them a nearly limitless supply of power as long as they can get access to hydrogen.
Then one day, without warning, the Earth is destroyed in such a way that it is no longer a viable source of any resources. Description of how I plan for the Earth to be destroyed is available for discussion here Apocalypse by Atomic Catalyst
Using only technology that could realistically be incorporated into space stations today (plus the 2 exceptions), could mankind use what is found in asteroids to survive indefinitely in space or would certain things simply be too difficult to supply, repair, or replace without Earth to continue the survival of the species?
Based on feedback, I see a there is infact a big hole in the basic principle of how long it would even take for this "near future" to happen; so, I'm gonna conclude my original model here as not worth passing the reality-check phase of development until I can get some key details worked out. Before I even begin to further speculate on this topic, I think I need to answer the more fundamental topic of What is the minimum timeframe required to colonize the asteroid belt?