 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2012/e/e3/b/ii/1/solution

Take

A_{1} = 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. Thus

A_{1}^{'} = R^{2} = (R^{2} ∖ A_{1})^{'} .

The infinitely-many-points observation ensures that one can avoid the centre of the disk itself in both cases.

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