[Kyrie]

But if you really want to be mathematical, why are you using metrics at all? Appending a point at infinity is unnatural in the theory of metric spaces, but perfectly reasonable in the theory of topological spaces. In this case "closer to infinity than" means "outside more--in the sense of containment of sets not cardinality--compact sets of 'finite points' than" in which case 10,000 is "closer to infinity" than 10 (lying outside the compact interval [0,99], for instance, while 10 is inside it).

Are you sure you can make this notion precise?

You might start by specifying the space you wish to use. I doubt you are thinking of the set of ordinals including omega, since the natural topology would be the discrete topology. So shall we agree that we are using the set of reals with an appended positive infinity?

Next, we need to specify a topology. Shall we agree that it is to be the "order" topology for our space? That is, the topology is the set of arbitrary unions of finite intersections of intervals of the form (a, b), where a and b are any two points in the space.

But now for your notion of closeness. Compactness does not help you here. Note that closed intervals are not the only compact sets in this topological space. There are infinitely many compact sets in this topological space that contain the points 1000 and infinity but do not contain 1,000,000. There are also infinitely many compact sets that contain 1,000,000 and infinity but not 1000.

Open neighborhoods do no better. We can observe the same thing for open sets as for compact sets.

Your notion of "closeness" seems to be based not on topology at all, but on intervals, i. e., order. Since 1000 < 1,000,000 < infinity, there are intervals containing 1,000,000 and infinity that exclude 1000 but not any that contain 1000 and infinity that exclude 1,000,000.

In this sense we can say that A is "closer" to C than B is to C. But we can't say how much closer it is. This is a weaker notion of "closeness" than we might like.

Notice what would happen if we were starting with the complex numbers rather than the real numbers. The Euclidean metric gives us a precise notion of "closeness"—until we append an infinity point. Then, as with the real numbers, we find that there is no translation-invariant metric that can be defined. However, there is no natural order on the complex numbers either. Then on what basis would we say that one complex number was "closer" to infinity than another?

BTW, it's nice to think about real mathematics again...thanks for the reply.

[Kyrie]

Did you, perchance, mean to say, "connected" instead of "compact"? You can make your argument that way, although on the extended real line it amounts to nothing more than the argument based on order.

[The_Reader_David]

You will note that I specified set containment not cardinality in by notion of "more". You are, however, correct that my formulation was inadequate to my point. The precise statement involves looking at the definition of +infinity as an end of the real line: an equivalence class of connected components of complements of compacta--which then, together with the formal point at +infinity, form a neighborhood basis of +infinity. The set of opens in this neighborhood basis which contain 10,000 is a proper subset of the set of opens in the neighborhood basis which contain 10, and 10,000 is thus "closer" to +infinity in this topological, non-metric sense than 10.