Let X be a compact set such that its compact open subsets form a basis for the topology. I ask if they form a $\sigma$-algebra. Let's denote this set by $\tau_c(X)$.
The first property is easy:
1)$U\in\tau_c(X)\Longrightarrow X\setminus U\in\tau_c(X)$
For the second property it is clear that the union is open. Therefore, my question reduces to ask if this set is compact.
If the answer is yes, I think it is key the fact that $X$ is compact.