Solution
= Solution
For each $n\in\mathbb N$, the open interval
$$
\left(n-\frac12,n+\frac12\right)
$$
meets $\mathbb N$ only in $\{n\}$. Thus every singleton is open in the <subspace topology> on $\mathbb N$. Every subset $A\subseteq\mathbb N$ is the union $A=\bigcup_{n\in A}\{n\}$ and is therefore open. Hence the induced topology is the <discrete topology>:
$$
\boxed{\mathcal T=\mathcal P(\mathbb N)}.
$$