I know that W. Rudin's book "Principles of Mathematical Analysis" discusses the extended real number system in P11-12, where two symbols

and

are introduced. But I think to extend the set

to include something standing for infinity, only one symbol need to be added, that is, the symbol

in the one-point compactification of

(See, e.g.

29 of Munkres' "Topology", this proccess makes the extended system compact, the added symbol

is a limit point of

, that's why you add a bar above

). So, both

and

are actually the same; there is no difference between them; we write + or - in front of

just in order to indicate how sequences in

converge to

-- whether the sequence is becoming larger and larger, or conversely. Am I right?