If $\mathcal{O}(X)$ is the collection of open sets in the topological space $X$ ordered by inclusion and $K\subseteq X$ is a compact subset of $X$ then it is easy to see that:
$$\mathscr{U}_K = \{O\in\mathcal{O}(X) : K\subseteq O\}$$
is open in the Scott topology on $\mathcal{O}(X)$. If $X$ is finite then the collection $\{\mathscr{U}_K:K\ \mbox{compact in}\ X\}$, is clearly a basis for this topology.
My question is: Is this collection always a basis for the Scott topology on $\mathcal{O}(X)$?