Building weight-free Følner sets for Yu's Property A in coarse geometry

Abstract: In this note we study the natural question of when the generalised F{\o}lner sets exhibiting property A can be chosen to be subsets of the space itself. We show that for many property A spaces $X$, this is indeed possible. Specifically this holds: for all discrete bounded geometry metric spaces which coarsely have all components unbounded; for all countable discrete groups; and for all box spaces.