The set of natural numbers N Is this open, closed, neither or both?



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The set of natural numbers N is closed.  We can by considering a real number `x in R` where `x>0` is not a natural number.  Now pick the natural number n so that n is the closest natural number below x.  Then pick a radius `r=min(x-n, n+1-x)` , and we see that we can create an open ball around the point x that doesn't touch any point n. Since we can repeat this process for each point n in N, we can form open balls that don't touch any points in N.  This shows that the complement of N is open, which means that N is closed.

N is a closed set. 

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