be the closed ball with radius $r$ centered at $x$. Let $X^{\lambda}$ be a Poisson point process on $M$ with intensity $\lambda$. Around every point of $X^{\lambda}$ we place a ball of unit radius, and denote by $C^{\lambda}$ the region of the space that is covered by some ball, that is $C^{\lambda}:=\cup_{x\in X^{\lambda}}S(x,1)$. We remark that all proofs below work if we instead consider the model with some arbitrary fixed radius $R$. Write $\P_{\lambda}$ for the probability measure corresponding to this model, which is called the Poisson Boolean model with intensity $\lambda$.

Next we introduce some additional notation. Let $V^{\lambda}:=(C^{\lambda})^c$ be the vacant region. Let $C^{\lambda}(x)$ be the component of $C^{\lambda}$ containing $x$. $C^{\lambda}(x)$ is defined to be the empty set if $x$ is not covered. Let $X^{\lambda}(A)$ be the Poisson points in the set $A$. Furthermore denote by $C^{\lambda}[A]$ the union of all balls centered within the set $A$. With $N_C$ and $N_V$ we denote the number of unbounded connected components of $C^{\lambda}$ and $V^{\lambda}$ respectively. The number of unbounded components for the Poisson Boolean model on a homogeneous space is an a.s. constant which equals $0$, $1$ or $\infty$. The proof of this is very similar to the 