My complex analysis textbook says the following:
A set $\Omega$ is bounded if there exists $M > 0$ such that $|z| < M$ whenever $z \in \Omega$. In other words, the set $\Omega$ is contained in some large disc.
Something about this definition doesn't seem right.
$|z| < M$ is an open disc centred at the origin, since an open disc centred at the point $z_0$ is defined as $D_r(z_0) = \{ z \in \mathbb{C} : |z - z_0| < r \}$; and so we have that a bounded set is $\{ z \in \Omega : |z| < M \} \subseteq \Omega$. In other words, the definition is saying that a set is bounded if every point in that set is in an open disc of radius $M$ centred at the origin. Am I correct in my understanding of this?
Therefore, this definition of a bounded set is specific to a disc centred at the origin.
Topologically speaking, I don't understand how this is a sensical definition for bounded sets? It seems to me that a definition of bounded sets needs to be more generalisable, rather than anchored to some arbitrary point, such as the origin.
I would greatly appreciate it if people could please take the time to clarify this.
Notice that if a set is contained in $D_r(z_0)$, then that set is contained in $D_{r+|z_0-z_1|}(z_1)$ for any $z_1$. In other words, the 'anchor' is irrelevant.