Describe the topology induced on the set $\mathbb N$ of positive integers by the euclidean topology on $\mathbb R$.
Let $n \in \mathbb N$ then we know $(n - \frac{1}{2}, n + \frac{1}{2})$ is open in the euclidean topology on $\mathbb R$. Now, by definition of relative topology, we know that $(n-\frac{1}{2}, n + \frac{1}{2})\cap\mathbb N = \{ n \}$ is open in the induced topology on $\mathbb N$ by the euclidean topology on $\mathbb N$. However, since $n$ was arbitrary, we thus have that every singleton set $\{ n \}$ is open in $\mathcal T _\mathbb N$. Also, since fro very $A \subseteq \mathbb N$, $A = \bigcup_{n \in A} \{ n \}$ is open, we have that $\mathcal T_\mathbb N$ is the discrete topology on $\mathbb N$.
Is this correct?
