TFF:"The implication, then, is that Inf and -Inf are now taken as actual points on the number line."
Yep.
TFF:"I'm not familiar with 'compactification.' Is this a way of circumventing the common notion of orderliness, usu we think something along the lines of given a point x and a positive c, there exist x-c and x+c, such that x - c < x < x + c, obviously not true when x=Inf.The "compactification" is a way of getting around this common idea?"
No, this is not the ideea. BTW, you have already introduced an additive group structure on the real line when you wrote the ordering relation the way you did above. Think points - not coordinates - and sets of points. And even in your case it works if you define Infty=supp{<R>}, -Infty=inf{<R>} with respect to this ordering relation, where <R> is a s previously the real line with the infties included.
The ideea of compactification is something along these lines: (0,1) is a non-compact interval while [0,1] is a compact interval (you have included the ends of the interval in the latter set. Same thing with the real line, you include the infties and you end up with the compactified real line.
