  We consider connectivity properties of the vacant set of (random) ensembles
of Wiener sausages in $\mathbb R^d$ in the transient dimensions $d \geq 3$. We
prove that the vacant set of Brownian interlacements contains at most one
infinite connected component almost surely. For finite ensembles of Wiener
sausages, we provide sharp polynomial bounds on the probability that their
vacant set contains at least $2$ connected components in microscopic balls. The
main proof ingredient is a sharp polynomial bound on the probability that
several Brownian motions visit jointly all hemiballs of the unit ball while
avoiding a slightly smaller ball.
