real analysis – Denumerable set – All points are boundary points

I am lookink for some examples on R for denumerable set that all points are boundary points.

Definition for boundary point (Bartle book): x is a boundary point of A if every neighborhood of x contains points in A and points in Ac

One example that I can intuitively think of are the irrational numbers, but I am not really sure how to formaly prove this.

Thank you