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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

Caprikuarius
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1 Answers1

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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.

J. W. Tanner
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Kavi Rama Murthy
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  • How did you show that III is correct? (both formally and intuitively are fine) – Caprikuarius Sep 03 '19 at 00:19
  • Continuity is equivalent to the fact that inverse image of any open set in $\mathbb R$ is open in $\mathbb Z^{+}$. This is obviously true here since every subset of $\mathbb Z^{+}$ is open. – Kavi Rama Murthy Sep 03 '19 at 00:24