$\mathbb{R}^\ast$, $\mathbb{R}_+$, $\mathbb{R}^\ast_+$ "deprecated"?

Question

I have recently stumbled upon this page in ProofWiki, which asserts that the notations $$ \mathbb{R}^\ast = \mathbb{R}\setminus \{0\} $$ and $$ \mathbb{R}_+ = \{x\in\mathbb{R} \mid x\geq 0\} $$ are both "deprecated." Is there a reason for that? (these notations are the ones I'm familiar with, and the ones I learnt since middle school)

Answer 1

If they're deprecated, they're deprecated for ProofWiki, certainly not in general. Any particular forum or journal is free to choose their own standards for mathematical notation. The main criterion should be to avoid confusion. The problem with $\mathbb R_+$, I think, is that it may not be obvious whether you mean $\ge 0$ or $> 0$. As for $\mathbb R^*$, perhaps it's that $^\star$ has many different meanings in different contexts.

Answer 2

There are still many people who use this notation. Still, I would recommend the following.

$\mathbb R^\times$ looks very much like $\mathbb R^\ast$, but makes it clear that you want the multiplicative ($\times$) elements of $\mathbb R$, i.e. $\mathbb R^\times = \mathbb R \backslash \{ 0 \}$, and not its dual $\mathbb R^\ast = \operatorname{Hom} (\mathbb R, \mathbb R) \cong \mathbb R$. In a more general situation, $R^\times$ (for any ring $R$) is used often for the set of units, and $R^\ast$ could mean "multiplicative elements" or "all elements except zero" or, again, the dual of $R$, i.e. $\operatorname{Hom} (R, R)$, thinking of $R$ as a module over itself.

$\mathbb R_+$ is less ambiguously written as either $\mathbb R_{>0}$ or $\mathbb R_{\geq 0}$. (Of course, paired with $\mathbb R_+^\ast$ the risk of confusion is not as great, since here $^\ast$ wouldn't mean the dual vector space, but then $\mathbb R_{\geq 0}$ is equally easy to write and unambiguous.)