I was thinking about if $\mathbb R$ could be written as $(-\infty , \infty)$. I'm not sure if it's okay, because I've read somewhere (I can't remember where) that $(-\infty , \infty)$ declares extended real line, which is totally different from $\mathbb R$.

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    $\begingroup$ Where you read that are more probably write...but it is a matter of agreement. Better, write $\;\Bbb R=(-\infty,\,\infty)\;$ . I don't think anyone dealing with mathematics will misunderstand that. $\endgroup$ – DonAntonio Aug 12 '18 at 8:01
  • $\begingroup$ Is it okay to write ??@DonAntonio $\endgroup$ – Anik Bhowmick Aug 12 '18 at 8:02
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    $\begingroup$ You can write it, its just more work and more to read. $\endgroup$ – copper.hat Aug 12 '18 at 8:04

Yes. Although it's unusual, it makes sense to write $\mathbb R$ as $(-\infty,\infty)$. That would never be the extended real line, which would be denoted by $[-\infty,\infty]$.

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  • $\begingroup$ But can $\infty$ be included in closed brackets ?? I mean, $\infty$ is not a number, how can it be an element of a set ?? $\endgroup$ – Anik Bhowmick Aug 12 '18 at 8:04
  • $\begingroup$ In extended real line, $\infty$ is a number (though it's not a real number). $\endgroup$ – user529760 Aug 12 '18 at 8:06
  • $\begingroup$ And $\emptyset$ is not a number either, but we write $\{\emptyset\}$, don't we? Besides, we are talking about the extended real line here, right?! Therefore, it is supposed to contain something else besides the real numbers. $\endgroup$ – José Carlos Santos Aug 12 '18 at 8:06
  • $\begingroup$ @JoséCarlosSantos: $\{\emptyset\}$ is really another case. That is the set containing the empty set, we can have sets of anything. $\endgroup$ – Henrik supports the community Aug 12 '18 at 8:29
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    $\begingroup$ @AnikBhowmick I'm glad I could help. $\endgroup$ – José Carlos Santos Aug 12 '18 at 11:31

It can be written that way, it doesn't mean it should be.

Have you ever heard of "complex infinity" or "directionless infinity"? Your readers would quickly realize you're not referring to either of those, but it could still be enough to cause a little cognitive dissonance and disrupt the flow of your presentation.

If you don't like $\mathbb R$ for whatever reason, you can always use $\textbf R$ instead. You don't lose any clarity that way.

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  • $\begingroup$ What's the difference between the two $R$'s? $\endgroup$ – goblin Aug 12 '18 at 22:30
  • $\begingroup$ @goblin Is this one of those you already know but you're checking if I know? $\endgroup$ – Robert Soupe Aug 13 '18 at 0:53
  • $\begingroup$ No I haven't heard any of them. And besides of that, what does $R$ denote ?? $\endgroup$ – Anik Bhowmick Aug 13 '18 at 1:48
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    $\begingroup$ $\mathbb R$ is blackboard bold for the set of reals, $\textbf R$ is for the set of reals for people who think blackboard bold is for the blackboard only, and $R$ can mean whatever you want it to mean just as long as you say what that is. $\endgroup$ – Robert Soupe Aug 13 '18 at 3:22
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    $\begingroup$ @RobertSoupe, fair enough. I thought maybe you were proposing that $\mathbf{R}$ be used for the affinely extended real line, or perhaps the projectively extended one. $\endgroup$ – goblin Aug 13 '18 at 8:53

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