Open/closeness of subsets of natural numbers

Question

So I've just started reading about neighbourhood and Hausdorff space. It makes me wonder if $(\mathbb{N},\mathcal{P}(\mathbb{N}))$ is Hausdorff and why, and are sets in $\mathbb{N}$ open or closed or?

Answer 1

Let $m,n\in\mathbb{N}$. Then $\{n\}$ and $\{m\}$ are in the power set and contain $m$ and $n$. But, they are disjoint. This demonstrates the Hausdorff property.

In fact, this holds for any discrete space.

Also, the open sets are by definition all subsets in $\mathbb{N}$. What are the closed sets? (Think about complements.)

Answer 2

Well, any topology of the form $(S,P(S))$ is the discrete space. So, it's basically a collection of countably many points, with no particular connections between them. This means that every set is open. Thus it is Hausdorff, since the singletons $\{a\}$ and $\{b\}$ are disjoint neighborhoods of any distinct $a$ and $b$ in the space.