TFF, the real line is another expression for the set of real numbers (there is a natural isomorphism between the set of real numbers and the points along an infinite straight line).
Now, if you think in terms of a line, an infinite line is a non-compact set as defined traditionally (does not include the end points at infinity). You can simply compactify it by including these points, and make the real line a closed interval.
Hm, references. Try something on modules, rings, fields,algebraic topology something like that. I will have to look for a more explicit reference.
But the ideea is simple to follow, algebraically speaking. Sure, it is a transfinite number. But you have to define how it behaves withe respect to the real numbers and the operations defined on the real numbers (infty +infty=infty, 1+infty=infty, infty*infty=infty, and so on and so forth). And the only problems you have are infty-infty, infty/infty, 0*infty and the similar. So you might as well treat it as a number, with these potential problems in mind.
