[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.

[Kyrie]

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.

You must have posted this while I was working (with distractions) on my previous post. All right, now you are giving a privileged position to your topological basis. But your basis for that topology is not inherent in your topological space, only in your construction of it. There are other bases for the same topology on the extended line. A different basis for the topology would give a different notion of "closeness." Then your notion of closeness is not from your topology, but from your basis; i.e., from your method of compactification. Isn't that a little bit arbitrary?