Check if $M = \{z \in \mathbb{C}| z = \frac {1}{n} + \frac {i}{m} \ with \ \ m,n \in \mathbb{Z} \backslash \{ 0 \} \} $ is compact

Question

I want to check, if this set is compact:

$M = \{z \in \mathbb{C}| z = \frac {1}{n} + \frac {i}{m} \ with \ \ m,n \in \mathbb{Z} \backslash \{ 0 \} \} $

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Thoughts: $z:= a +bi$

• real part $a$ is $\frac {1}{n}$ so I think it is bounded because $a \in [-1,1]$
• imaginary part $b$ is $\frac {i}{m}=\frac {1}{m} * i$ so I think it is also bounded because $b \in [-i,i]$
• => both parts are bounded => $z$ is bounded too

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1. Are my thougts correct?
2. How can I check if $M$ is closed?

Answer 1

Ok for $M$ bounded. But an argument like: $$\forall z\in M, z\in \overline B(0,1)=\{z\in\mathbb Z\mid |z|\leq1\}$$ would have been better. But anyway, you proved that $M\subset [-1,1]\times [-1,1]$ which is good too.

But your set is not compact because $0\notin M$. Indeed, $0\in \overline M$ but not in $M$.