Are the closed and unbounded subsets of $\mathbb{R}$ known up to homeomorphism? I am currently working on a problem for which this knowledge could greatly reduce the number of cases, but I have yet to find anything after searching online. Are the closed unbounded subsets of $\mathbb{R}$ known up to homeomorphism?
References would be greatly appreciated.
 A: For any closed subset $K$ of the usual Cantor set $\mathcal{C} \subseteq [0,1]$, consider the map which sends $K$ to the set $K \cup [2,\infty)$. Then this map explicitly reduces the problem of classifying closed subsets of the Cantor set up to homeomorphism to the problem of classifying closed non-compact subsets of $\mathbb{R}$ up to homeomorphism.
The classification problem for closed subsets of a Cantor space up to homeomorphism is known to be pretty complicated. More specifically, it is as complicated as the classification problem for countable graphs, or say, countable linear orders up to isomorphism. (These claims can be made precise using the notion of Borel reducibility. You can find the relevant theorems in this paper. But I doubt that you are looking for such technical details.)
A: Infinite, complete, separable linear order with at most countably many jumps and not both a greatest and least element.
See 
http://www.math.uni-hamburg.de/home/geschke/papers/SeparableLinearOrders2.pdf
