Determine if the set $\{(x, \alpha, \beta) \mid x \ge 0 \land 0 \le \alpha \le \pi \land 0 \le \beta \le 2 \pi\}$ is compact

Question

Determine if the set $$\{(x, \alpha, \beta) \mid x \ge 0 \land 0 \le \alpha \le \pi, 0 \le \beta \le 2 \pi\}$$ is compact

I tried to solve this right from the definition of a compact set. A set is compact if and only if it is closed and bounded.
I guess that this set can be represented by an "infinitely long" cuboid - namely, with the sides of $x$, $\pi$ and $2 \pi$. If this set were bounded, we would be able to find a sphere centered at the origin containing the whole set. Now, imagine that such sphere does exist and that its radius is $R$. This point: $(R+1, 0, 0)$ is a member of the set in question but is outside the sphere.
Therefore, the set is not bounded.
Since it is not bounded, it is not compact.

Please, excuse me if this problem was too simple, but I have only just started my topology course and this is the very first problem I have come across - I'm willing to understand the basics very well.

Is my solution to the problem correct?

Answer 1

$\text{ }$ Yes, everything correct. $\text{ }$