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Is there any standard symbol for the set $\{x\in\mathbb{R} : x > 0\}$?

I think $\mathbb{R}^{+}$ usually includes zero. Some sources say I should use $\mathbb{R}^{*}_{+}$ but it looks slightly bizarre to me.

Suggestions?

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  • $\begingroup$ I've seen $\mathbb R_{++} $ for the positive real numbers. $\endgroup$
    – littleO
    Jul 19 '15 at 9:04
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$\mathbb{R}_{\ge a}$ is VERY standard for $[a,+\infty)\subset\mathbb{R}$ and $\mathbb{R}_{> a}$ for $(a,+\infty)\subset\mathbb{R}$

This is a very obvious "There is no other sensible interpretation" convetion, if I give you $\mathbb{R}_{\le -5}$ you know immediately I mean $(-\infty,-5]$

I have seen $\mathbb{R}^+$ used - this follows the $\mathbb{N}^+=\{1,2,\cdots\}$ convention but I don't like this because it isn't as obvious.

Oh to answer your question, strictly positive reals: $$\mathbb{R}_{>0}=(0,\infty)=\{x\in\mathbb{R}|x>0\}$$

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There is no one single universal standard symbol recognised and used by everyone. Something like $\mathbb{R}^{>0}$ or $\mathbb{R}_{>0}$ is clear enough (I have seen people use both); $\mathbb{R}^*_+$ makes sense but I've never seen anyone use it.

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    $\begingroup$ $\Bbb{R}^*_+$ is used in french books and francophone countries. It's the set of invertible positive real numbers $\endgroup$
    – user5402
    Jul 19 '15 at 14:53
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Well, why don't you give $(0,+\infty)$ a chance? Anyway, positive numbers are different from non-negative numbers: therefore I'd guess that $\mathbb{R}^{+} = \{ x \in \mathbb{R} \mid x>0\}$, and $\{ x \in \mathbb{R} \mid x \geq 0\} = \mathbb{R}^{+} \cup \{0\}$. But many people prefer to include zero for mere convenience, so that it is improbable to reach a universal agreement.

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    $\begingroup$ $(0,+\infty)$ is denoted $]0;~+\infty[$ in a lot of countries. It makes more sense. $\endgroup$
    – user5402
    Jul 19 '15 at 14:57
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    $\begingroup$ @metacompactness I've always hated reversed brackets. To me $($ means open, $[$ means closed. By the way, here in Italy many colleagues follow your notation, but I learned analysis from Rudin's books. $\endgroup$
    – Siminore
    Jul 19 '15 at 16:51
  • $\begingroup$ @Siminore The notation $(a,b)$ is inconsistent with geometric notation. For instance, $[AB]$ means the segment including $A$ and $B$; $]AB[=[AB]-\{A;~B\}$ whereas $(AB)$ is a line and not a segment. $\endgroup$
    – user5402
    Jul 19 '15 at 19:38
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    $\begingroup$ @metacompactness never seen your notation. For me a segment is $\overline{AB}$. But wait, I am an analyst, I write $[u,v]$ for any segment in any normed space! :-) $\endgroup$
    – Siminore
    Jul 20 '15 at 8:29
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    $\begingroup$ @metacompactness I never felt such a need in my life. If it should happen, I'd clarify by words. I mean, if you are an analyst, you use intervals every minute and never use segments. I suspect that the opposite thing happens if you are a geometer. $\endgroup$
    – Siminore
    Jul 20 '15 at 12:16

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