Definition of Domain and Region

Question

First of all, I would like to ascertain the definition of domain vs region.

Definition (Region)
$\Omega \subset \mathbb{R}^{N}$ is a region if and only if its interior is nonempty.

Definition (Bounded Region)
$\Omega \subset \mathbb{R}^{N}$ is a bounded region if and only if it is a region and $\exists R>0, \, \Omega \subset B(0,R)$.

My question is how to show that $\Omega = \mathbb{N}$ is not a region?

1 Do I have a correct definition of region?
2 Is the word "region" and "domain interchangable?
3 In the context of a domain for partial differential equation, do we say domain as "domain of a function" or a "region" instead?

Any help is much appreciated!

Answer 1

You can observe that for any natural point $n\in \mathbb{N}$ you have that for each $\epsilon>0$ then

$B(n,\epsilon) \cap \mathbb{N}^c\neq \emptyset$

So $n$ cannot be a internal point of $\mathbb{N}$