Given some topological space $\mathbf{X}$, we consider the Fell topology on the set of closed subsets of $\mathbf{X}$. This is generated by sets of the form $I_U = \{A \mid A \cap U \neq \emptyset\}$ with $U$ ranging over open subsets of $\mathbf{X}$ together with sets of the form $D_K = \{A \mid A \cap K = \emptyset\}$ where $K$ ranges over compact subsets of $\mathbf{X}$. Let $\mathcal{F}(\mathbf{X})$ be the topological space constructed as such.
It seems to be known that if $\mathbf{X}$ is locally compact, then $\mathcal{F}(\mathbf{X})$ is compact.
What is known about the inverse implication? I am interested in both the general case, and the restriction to countably-based based spaces.
Bonus question: What exactly is meant by "local compactness" here?