What are the limit points of Q in the metric space (R,d)?

[natnat_nuts]

Homework Statement

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.

Homework Equations

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.

 

[micromass]

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.

This reasoning is based on that the open intervals are the balls of the metric. And this is true if the metric is d(x,y)=|x-y|.

But here, the balls are a lot smaller. For example: B(x,1/2)={x}, so the singletons are open sets. What does that mean for the limit points?

 

[natnat_nuts]

does that mean that the set of limit points of A is {} null set since the singletons do not have other elements other than x?

 

[micromass]

does that mean that the set of limit points of A is {} null set since the singletons do not have other elements other than x?

Yes, it would be the null set. All points in $\mathbb{R}$ are isolated points with that metric.