what is the interior and the closure of $\{4\}$ and of $\{2,4,6,8.....\}$? let $n \in \mathbb{N}$ . Set $A_n = \{ n , n+1, n+2,....\} $ and $ T = \{ \emptyset , A_n\}_{ n\in \mathbb{N} }$
$1.$Is  $T $is separated ?
$2.$ what are dense  set in $T$ ?
$3.$ what is the interior and the  closure of $\{4\}$  and of $\{2,4,6,8.....\}$?
My attempt :  for  $ 1,$  I know  that $ T$ is not separated because  take $A_1 = \{ 1, 2,3,...\}$ and $A_2 = \{ 2, 3,4,,,,,,,\}$ here both $A_1 \cap A_2 \ne \emptyset$
For  $2)$ i thinks  $\mathbb{N} $
For $3)$ im confused
Any hints/solution will be appreciated
thanks u

Your answers:
1) Incorrect. The fact that two particular open sets intersect does not imply at all that the topology is not separated. You should pick two natural numbers $m,n$ (call $m$ the lesser one, for example) and try to show that this points can not be separated.
2) Incorrect. Of course $\Bbb N$ is dense. But how are the closed sets? What would be the closure of the set of, say, even numbers?
For the third one, perhaps you have a clearer idea of how the topology is. Good try!
