  In this work we reproduce the characterization of $\Gg^s$-sets from the
euclidean setting [J. London Math. Soc. 49:267-280,1994] to more general metric
spaces. These sets have Hausdorff dimension at least $s$ and are closed by
countable intersections, which is particularly useful to estimate the dimension
of the so called sets of $\alpha$-approximable points (that typically appear in
Diophantine approximations).
