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?



Compactness of the Fell topology and local compactness
Local compactness means that every point has a compact neighborhood. That is, every point is interior to a compact set. The Fell topology on the nonempty closed sets of a Hausdorff space is always compact, without any further assumption. However, local compactness is equivalent to the Fell topology being Hausdorff.


You can find a lot more material on the topic in the 1993 book "Topologies on closed and closed convex sets" by Gerald Beer.


