The question is: let $\mathbb{Z}^+$ be the set of positive integers and let d be the metric on $\mathbb{Z}^+$ defined by $d(m, n) = \begin{cases} 0\text{ if }m = n\\ 1 \text{ if } m \neq n \end{cases}$ for all $m, n \in \mathbb{Z}^+$. Which of the following statements are true about the metric space $(\mathbb{Z}^+, d)$?
I. If $n \in \mathbb{Z}^+$, then {n} is an open subset of $\mathbb{Z}^+$.
II. Every subset of $\mathbb{Z}^+$ is closed.
III. Every real-valued function defined on $\mathbb{Z}^+$ is continuous.
The answer says that all I, II, and III are true. However, I have no clue for how to approach this problem. Could anyone tell me how does the metric space can impact these three statements? (To me, it seems that the true/false of these 3 statements have nothing to do with the metric space).
Thanks
This is the so-called discrete metric, and the assertions are true even if you replace $\mathbb Z^{+}$ by any other set. The reason: the open ball of radius $1$ around any point $n$ is $\{n\}$. Hence every singleton set $\{n\}$ is open. Since unions of open sets are open, it follows that every subset is open.