Notation for the cartesian plane without the origin

Question

Its common to define $\mathbb{R}^{*}$ as the set of real number without zero: $\mathbb{R}^{*} = \mathbb{R} - \{ 0 \}$. But what is the notation for $\mathbb R^2$ without the origin $(0,0)$? What about for $\mathbb R^n$ without the origin?

Answer 1

I don't believe there is any standard notation for this (other than "$\mathbb{R}^2\setminus\{(0,0)\}$" of course). In particular, I would understand $(\mathbb{R}^2)^*$ to refer (if not to the dual space!) to the set of points in the plane with both points nonzero, that is, the set of invertible elements of $\mathbb{R}^2$.

That said, I would also use "$-^\times$" instead of "$-^*$" for this, to avoid confusion with the dual space as mentioned above; and I also recommend that for the one-dimensional setting.

Answer 2

I like to use $ \dot A $ or $ A ^ { \textstyle \cdot } $, where $ A $ is a set with a special element $ \alpha $, to denote $ A \setminus \{ \alpha \} $. So in particular, I write $ \dot { \mathbb R } $ for $ \mathbb R \setminus \{ 0 \} $ and $ \dot { \mathbb R ^ 2 } $ (or better $ ( \mathbb R ^ 2 ) ^ { \textstyle \cdot } $ to avoid confusion with $ ( \dot { \mathbb R } ) ^ 2 $) for $ \mathbb R ^ 2 \setminus \{ ( 0 , 0 ) \} $. This is hardly standard, and the special point must be specified anyway, so you should explain this when you use it. But at least it doesn't conflict with things like $ V ^ * $ for the dual of a vector space $ V $ or $ R ^ \times $ for the group of invertible elements of a ring $ R $.

Answer 3

You can consider punctured (deleted) neighborhood of $(0,0)$ when whole $\mathbb{R^2}$ is looked as neighborhood. There are many notations for it:

1. Classical: $\mathring{U}(x_0)$ on page 236 in Vladimir A. Zorich - Mathematical Analysis I-Springer, 2016 where $U(x_0)$ is neighborhood for $x_0$
2. Original: $G_{\neg p}(r)$ on page 221 in Zakon E. - Mathematical analysis 1-Trillia Group, 2004 $G_{\neg p}(r)=G(p,r)\setminus \{p\}$, where $G$ is spherical neighborhood with radius $r$.

Answer 4

You could write $\mathbb R^2\setminus\{0\}$, and your audience would almost certainly understand that the "$0$" in this notation stands for the neutral element of the vector space $\mathbb R^2$, i.e. the point $(0,0)$. This is slightly shorter than writing $\mathbb R^2\setminus\{(0,0)\}$. Similarly, you can write $\mathbb R^n\setminus\{0\}$ for $\mathbb R^n$ minus the origin, and no one would raise an eyebrow.

I don't think there is any widely accepted notation that is shorter than $\mathbb R^2\setminus\{0\}$, and so if you are going to use shorter notation, then you ought to define it first.