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**zzzhhh** I know that W. Rudin's book "Principles of Mathematical Analysis" discusses the extended real number system in P11-12, where two symbols $\displaystyle +\infty$ and $\displaystyle -\infty$ are introduced. But I think to extend the set $\displaystyle \mathbb{R}$ to include something standing for infinity, only one symbol need to be added, that is, the symbol $\displaystyle \infty$ in the one-point compactification of $\displaystyle \mathbb{R}$(See, e.g. $\displaystyle \S$29 of Munkres' "Topology", this proccess makes the extended system compact, the added symbol $\displaystyle \infty$ is a limit point of $\displaystyle \mathbb{R}$, that's why you add a bar above $\displaystyle \mathbb{R}$). So, both $\displaystyle +\infty$ and $\displaystyle -\infty$ are actually the same; there is no difference between them; we write + or - in front of $\displaystyle \infty$ just in order to indicate how sequences in $\displaystyle \mathbb{R}$ converge to $\displaystyle \infty$ -- whether the sequence is becoming larger and larger, or conversely. Am I right?