Is something like the following on the right track?

ro = ResourceObject["General Topology EntityStore"];
store = ResourceData[ro, "EntityStore"];
PersistentValue["GeneralTopologyEntityStore"] = store;

Then

Quit[]

in a new session:

AppendTo[$EntityStores, PersistentValue["GeneralTopologyEntityStore"]];

RandomEntity["GeneralTopologyConcept"]
(* Entity["GeneralTopologyConcept", "SpaceWithTopologyOfPointwiseConvergence"]*)

@Murray Eisenberg I would like to know how was the dataset downloaded to the local system. Could you please help me with this? Thank you

I don't believe there is a way to make the new store permanent, I think at startup the $EntityStores is empty by design.

But you can add the relevant code to your .init file, where you can keep any code you want executed when the kernel starts. I have tons of stuff in my init file, little helper functions I find to be useful.

To find the init file, run

SystemOpen@Echo @ FileNameJoin[{$UserBaseDirectory, "Kernel", "init.m"}]

add the lines

store = Get[ "~/Downloads/General-Topology-EntityStore.wl" ]; 
AppendTo[ $EntityStores , store]; 

And then when you restart your kernel you should be able to run

EntityValue["GeneralTopologyTheorem", "Activate"] // Activate;
Entity["GeneralTopologyTheorem", "ClassicalAscolisTheorem"]["QualifyingObjects"]

and get the result. There isn't a way to have the "Activate" code run beforehand, since the EntityValue framework is loaded after the init.m file is read.

@Murray Eisenberg First you need to figure out why you get $Failed when trying to Get the file (<<).

Do you get True from FileExistsQ["~/Downloads/General-Topology-EntityStore.wl"]? What happens when you try to use Import instead of Get?

As to the persistence - I think that is just the way $EntityStores was designed. If you solve the problem above and are able to read in the store using Get, then you could add the line AppendTo code above to your "~/Library/Mathematica/Kernel/init.m" file and it would be available via EntityValue every time you start a kernel

If you've already downloaded the file "General-Topology-EntityStore.wl" linked to from here, then all you need to do in your notebook is run this command

store = << "~/Downloads/General-Topology-EntityStore.wl"; (* or wherever you downloaded the data to *)
AppendTo[$EntityStores, << store]

and then you can access all the data via EntityValue