  We show that limsup sets generated by a sequence of open sets in compact
Ahlfors $s$-regular space $(X,\mathscr{B},\mu,\rho)$ belong to the classes of
sets with large intersections with index $\lambda$, denoted by
$\mathcal{G}^{\lambda}(X)$, under some conditions. In particular, this provides
a lower bound on Hausdorff dimension of such sets. These results are applied to
obtain that limsup random fractals with indices $\gamma_2$ and $\delta$ belong
to $\mathcal{G}^{s-\delta-\gamma_2}(X)$ almost surely, and random covering sets
with exponentially mixing property belong to $\mathcal{G}^{s_0}(X)$ almost
surely, where $s_0$ equals to the corresponding Hausdorff dimension of covering
sets almost surely. We also investigate the large intersection property of
limsup sets generated by rectangles in metric space.
