Consider the following set
$$A = \left\{(x, y, z) ∈ R^3: (x + z)^4 + (cos(y + z))^3 < 0\right\}$$
and answer the following questions (justifying your answers):
- Is A an open set?
- Is A a bounded set?
How would I solve these? I know that if $A$ is an open set and $f$ is a continuous function, then $f^{-1}(A)$ is open, and that we can apply that to show a set is open, but I'm not sure how. A hint would be great.
Consider the map$$\begin{array}{rccc}f\colon&\mathbb R^3&\longrightarrow&\mathbb R\\&(x,y,z)&\mapsto&(x+z)^4+\cos^3(y+z).\end{array}$$It is continuous and$$A=f^{-1}\bigl((-\infty,0)\bigr).$$Therefore, $A$ is open. And it is unbounded, since$$A\supset\left\{\left(-2n\pi+\frac\pi2,0,2n\pi-\frac\pi2\right)\,\middle|\,n\in\mathbb N\right\}.$$