Let $S$ be a set, or equivalently the topological space with discrete topology and $2$ two point set with discrete tiopology. $βS$ be the Stone–Čech compactification of $S$. By Tychonoff theorem the topology on $2^S$ is compact with respect to the product topology.  
Is the compact-open topology on $2^{\beta S}$ the product topology or it is something much stronger, say uniform topology? How one can see it.