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.

In particular can $2^{\beta S}$ be a compact in case $S$ is an infinite set.

Given X, $2^X$ means the topological space with following property:

- There exists continuous map, an evaluation map $e:2^{X}\times X\to 2$
- For any compact Hausdorff and zero-dimensional topological space A and continuous map $f:A\times X\to 2$, there exists $\hat{f}:A\to 2^{X}$ such that $f=e\circ (\hat{f}\times id_{X})$. The last says that if $A=\{*\}$ a one point set then $2^X$ as a set is set of continuous maps from $X$ to $2$.