# Compactness of the Fell topology and local compactness

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?

• It's the first time that I hear that the above topology is called Fell topology (sometimes it was called Hausdorff topology, with the understanding that it was a generalized Hausdorff topology. Could you say more about the history of this topology? – Wlod AA Jun 23 '18 at 21:48
• – Adam Epstein Jun 24 '18 at 7:58
• @AdamEpstein, thank you. Was Fell already born when they invented the topology under consideration? (My classical association of "fell" and topology is the horrible Hurewicz's fall in Mexico which caused him his life). – Wlod AA Jun 24 '18 at 9:02