Need some help with Examples 2.15 and 17. I think in 2.15, apart from the sets formed by 1st 3 elements {{0},{1},{2}}, everything else should be in T_d for this to be a topology on N. Same kind of justification I have for 2.17. However, I might be wrong so please check my answer and correct me if I am wrong.
Any help will be greatly appreciated. Thanks!!
All subsets $A$ of $\Bbb N$ are unions of singletons: $A = \bigcup \{\{n\}: n \in A\}$ so the topology contains all subsets of $\Bbb N$ if it contains the singletons. So $\mathcal{T}_d=\mathscr{P}(\Bbb N)$
2.17 contains all unions of those doubletons (they are disjoint, so intersections add nothing new). In fact if you think about it for a while the topology is given in total by $O \in \mathcal{T}_{dd}$ iff $\forall n \in \Bbb N: (2n \in O) \iff (2n+1 \in O)$.