I have an elementary question regarding this definition, from the site of the University of Bath, UK:
Theorem 2.18. $C^k(\Omega)$ is Banach. Let $\Omega \subset \mathbb{R}^N$ be a bounded open set and let $k \in \mathbb{N}$. Then $C^k(\bar{\Omega})$ is a Banach space with the norm
\begin{equation*} {||f||}_{C^k(\Omega)} = \sum_{|\alpha| \le k} \sup_\Omega {|\partial^\alpha f|}\text{,} \end{equation*}
where here the sum is over all multiindices $\alpha$ with order $\leq k.$
Why is it important to consider $\Omega$ to be a bounded set? (In case of euclidean space, maybe boundedness would help us when considering the support of a function, but I am finding it difficult to understand the importance of boundedness, in general.)