Which of these points in my set are limit points of that set?

Question

Let $X=\{(x,y)\in \mathbb{R}^2| x,y\in\mathbb{Q}\}$. Which points in the plane are limit points of $X$?

I say there are no limit points in this set since $\exists\epsilon>0$ such that $B_{\epsilon}(x)$ does not meet $X$ in another point. This results because there are an infinite number of values between rational and irrational numbers.

Is my reasoning correct?

Answer 1

Your statement is not correct. If you think about the corresponding problem in one dimension, you claim the rationals have no limit points. This is inconsistent with the fact that the rationals are dense in the reals. Given any real $x$ and a radius $\epsilon$ there are rationals in the interval $(x-\epsilon, x+\epsilon)$