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?

Fell topology(sometimes it was calledHausdorff topology, with the understanding that it was ageneralized Hausdorff topology. Could you say more about the history of this topology? $\endgroup$ – Wlod AA Jun 23 '18 at 21:48