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Consider the topological space $\mathbb{R}$ equipped with the standard topology. Determine the interior, closure and boundary of this set and decide whether it is open or closed:

$a \in \mathbb R: K_a = \{ \frac{a}{n+m} | n, m \in \mathbb N_+ \} $

So, I know that all of the elements of this set lie somewhere in the intervall $]0; \frac a 2]$ for $ a > 0$ or in $]\frac a 2; 0]$ for $a < 0$. I also know, that the elements are not continuous, i.e. it is a pointwise set. But at this point I am finished and do not know, how to carry on:

I don't understand how I can determine the 3 properties of the set (interior, closure and boundary), when the set is made of numbers that are not neighboring (if you understand what I mean)

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up vote 1 down vote accepted

Surely you have $K_a=\{\frac{a}{n+1}|n \in \mathbb{N}\}=\{\frac{a}{2},\frac{a}{3},\frac{a}{4},\ldots\}$

For an element to be in the interior you would need an open $\epsilon$ neightborhood of the elment to be in the set which is not the case here, as you only have isolated points. The closure of the set is the smallest superset that is closed under the convergence of sequences, so you would have to add the element $0$ because you can have sequences that converge against $0$.

Now the boundary is the closure without the interior and there you have all your sets.

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