I'm having a hard time understanding why the rationals are not locally compact. The definition of local compactness is that every neighborhood of x in X is contained in some compact subset of X. But what are the compact subsets of X? I think this is my biggest problem. I know that the compact subsets of the reals are just the closed and bounded intervals, but I am unsure as to what they look like in the space of rationals. Could i say that in any interval about x, there contains both rationals and irrationals (the set of rationals and irrationals is dense in the reals), so any compact set would have to contain irrationals too... or something :(. I'm very confused. Thanks for the help.