I am looking for a way to notate the set in $\mathbb{R}^3$ given as $\Big\{ (x,y,z): z = 0 \Big\}$, ie, the $xy$-plane (or the plane $z=0$). I want a notation which may also work with notating the $z\geq 0$ half-space. Some ideas I had are:
- $Z_0$ and $Z_0^+$ to indicate the plane and the half-space respectively but this sits a bit unwell in my stomach.
- Simply $Z$ and $Z^+$ but this seems too ambiguous.
- $XY$ for the plane and $\mathbb{R}^3_+$ for the half-space but then there is no relation between the two and this bothers me a little.
What would you pick to notate them?
Well, you may introduce the projections $\pi_i: {\Bbb R}^3 \to {\Bbb R}$ defined by $\pi(x_1, x_2, x_3) = x_i$. Then the $xy$-plane is $\pi_3^{-1}(0)$ and the half-plane is $\pi_3^{-1}([0, + \infty])$. You may also use the following notation: $$ P_0 = \{(x,y,z) \in {\Bbb R}^3 \mid z = 0\} \quad \text{and}\quad P_{\geqslant 0} = \{(x,y,z) \in {\Bbb R}^3 \mid z \geqslant 0\}\ $$