(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I am not sure if I should post it under Calculus and Analysis but since it is for my Advanced Calculus class the I decided to do it anyways.

If the metric space (S,d) is defined as S = R (set of real numbers) and d(x,y) = 0 if x=y and d(x,y) = 1 if x is not equal to y, find the limit points of A if A = Q (set of rational numbers). Based on my notes, the answer is Q but I don't seem to get it.

Note: a point x element of S is a limit point of A if every open ball B(x,r) for r>0 (that is, an open ball or open interval that has center x and radius r) contains a point y element of A other than x.

2. Relevant equations

3. The attempt at a solution

My answer is either {} null set or R. If I take a rational number x to be the center of an open interval, by property of real numbers, I can always find at least one rational number not equal to x that is within the open interval no matter how small or big the interval is. Therefore, all rational numbers are limit points and they are in the distance of 1 from each other provided that they are not equal. But following this logic also means that if the center is an irrational number say y, then I can always find a rational number within the open interval no matter how small or big the interval is. Therefore, R is the derived set or the set of limit points.

On the other hand, I am also thinking that if the distance is 1 if the numbers are not equal and 0 if they are equal mean that all Q are isolated points and therefore the set of limit points is null set.

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# Finding the limit points

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