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Let $\mathbb{Z}^+$ be the set of positive integers and let $d$ be the metric on $\mathbb{Z}^{+}$ defined by

$$ d(n,m) = \begin{cases} 0 &,\text{ if }\; m=n \\ 1 &, \text{ if }\; m \ne n \end{cases} $$

for all $m, n \in \mathbb{Z}^+$ which of the following are true about the metric space $(\mathbb{Z}^+, d)$?

  1. If $n \in \mathbb{Z}^+$ then $\{ n \}$ is open.

  2. Every subset of $\mathbb{Z}^+$ is closed.

  3. Every real valued function defined on $\mathbb{Z}^+$ is continuous.

My thoughts. $\{n\}$ is open since it is the ball $B(n,1)=\{m\in \mathbb{Z}^+ : d(n,m)<1\}=\{n\}$ so it is the discrete topology on $\mathbb{Z}^+$. Hence the first and the second and the third are correct?

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    Correct. The topology induced by $d$ is exactly the discrete topology. – 2017-02-20
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    @Hermès how can I sign this question as solved? – 2017-02-23
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    @Hermès too humble to post as answer? :P – 2017-02-26
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    You should answer your own question in this cases. And should make it most explicit as possible on how you did your calculations – 2017-03-18

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