Past exam of the options II course of the University of Oxford / 2012 / E / E3 / b / ii / 1 / Solution

Take $A_1={\mathbb{Q}}^2$. Every open disk contains infinitely many points with both coordinates rational, so every point of the plane is an accumulation point of $A_1$. Every disk also contains infinitely many points with an irrational first coordinate, and these lie in the complement. ThusThe infinitely-many-points observation ensures that one can avoid the centre of the disk itself in both cases.