  We examine the question whether random set attractors for continuous-time
random dynamical systems on a connected state space are connected. In the
deterministic case, these attractors are known to be connected. In the
probabilistic setup, however, connectedness has only been shown under stronger
connectedness assumptions on the state space. Under a weak continuity condition
on the random dynamical system we prove connectedness of the pullback attractor
on a connected space. Additionally, we provide an example of a weak random set
attractor of a random dynamical system with even more restrictive continuity
assumptions on an even path-connected space which even attracts all bounded
sets and which is not connected. On the way to proving connectedness of a
pullback attractor we prove a lemma which may be of independent interest and
which holds without the assumption that the state space is connected. It states
that even though pullback convergence to the attractor allows for exceptional
nullsets which may depend on the compact set, these nullsets can be chosen
independently of the compact set (which is clear for $\sigma$-compact spaces
but not at all clear for spaces which are not $\sigma$-compact).
