  Global random attractors and random point attractors for random dynamical
systems have been studied for several decades. Here we introduce two
intermediate concepts: $\Delta$-attractors are characterized by attracting all
deterministic compact sets of Hausdorff dimension at most $\Delta$, where
$\Delta$ is a non-negative number, while cc-attractors attract all countable
compact sets. We provide two examples showing that a given random dynamical
system may have various different $\Delta$-attractors for different values of
$\Delta$. It seems that both concepts are new even in the context of
deterministic dynamical systems.
