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Decide the closedness and openness of {x:0<|x|<1 and (1/x)∉N}. I know the answer that the set is open but not closed. But I am not getting why?

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Your set $A$ is open because it is equal to union of intervals $$ ( \frac{1}{n} ; \frac{1}{n + 1} )$$ and $A$ is not closed because $$ 0 \in \mathbb{R} - A $$ but every interval $$ ( -\varepsilon ; \varepsilon)$$ have nonempty intersection with $A$

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Open is trivial. And it is not closed because the number in form of n^{-1} is limit point, and not belong to the set of problem.

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    Saying "your question is trivial" does little to answer the question. That which you did explain is insufficient for the kind of student who would ask this question – 2017-02-15