Exterior points in topology

Question

Consider the topology $\tau$ on $\mathbb{N}$ which consists of $\varnothing$ and all subsets of $\mathbb{N}$ of the form $\{G:\,G ⊆\mathbb{N}\text{ and }\mathbb{N}\setminus\Bbb G\text{ is finite}\}$. Then

Find the exterior set of the set $A=\{2,4,6,8,...120\}$.

Sol: $\text{Ext}(A)=\{21,22,23,...\}$

Is this correct?

Answer 1

If $O\subset A$ is open, then $\mathbb N\setminus O$ is finite. But $\mathbb N\setminus O\supset\mathbb N\setminus A\supset \{1,3,5,7,\dots \}$. Hence $\mathbb N\setminus O$ is infinite... Thus the only open set in $A$ is $\emptyset$.

Thus the exterior of $A$ is the whole space $\mathbb N$.