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Our teacher asked to give an example of set whose interior is countably infinite. I can't construct it.

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    a subset of $\mathbb R$? – 2017-01-28
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    An interior is always an open set. In $\Bbb R^n$ all the nonempty open sets are uncountable. – 2017-01-28

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In the metric space $\mathbb R$ every open set is either empty or uncountable. This is because open sets are unions of open intervals, and the non-empty open intervals are all uncountable. So no subset of $\mathbb R$ is countably infinite.

However there are some other metric spaces in which countably infinite open sets exist. It suffices to take a metric space that is itself countably infinfite. An example of such a metric space is $\mathbb Z$ with the euclidean norm. Notice that in this metric space $\mathbb Z$ itself is an open set that is countably infinite (so its interior is itself and is countably infinite).

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    Ok but how do I show that in Z , Z is open? – 2017-01-28
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    @JorgeFernándezHidalgo: much clearer! – 2017-01-28
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    the complete set is always an open subset of itself, for every metric space. It is clearly the union of all of the open balls. – 2017-01-28
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    Thank you !! You explained it beautifully – 2017-01-29
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    No problem ${}{}{}$ – 2017-01-29
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In $R$ under the usual metric it can't be since if it is an interior then it is open and if it is open then it should contain a nbd of each of it's points and nbds in$ R $are uncountable.and a countable set cant contain an uncountable subset