I am solving problem 12 (chapter 1) in the book Computational topology for DA by Krishna Dey & Wang.
I have a question about notation. It is asked to prove that $f^{(-\infty,0]}\cap S^{2}$ and $f^{[0,\infty)}\cap S^{2}$ are homotopy equivalent to two points and $S^{1}$, respectively. Here $f(x,y,z)=3x^{2}+3y^{2}+9z^{2}$ for all $x,y,z \in \mathbb{R}$, and $S^{i}$ is the $i-$sphere.
I thought $f^{(-\infty,0]}$ and $f^{[0,\infty)}$ where the inverse images of $(-\infty,0]$ and $[0,\infty)$ under $f$. However, this would not make sense, as these sets are $\{(0,0,0)\}$ and $\mathbb{R}^{3}$, respectively.
I looked for this notation in Chapter 1, but I truly couldn't find it. Does somebody know what it means?
Thank you!