This is my first question in this site, so please don't bo too hostile to me. If I made any mistakes, please let me know.
In my story, there is an island inhabited by people, numbering enough to sustain reproduction. There are continents (and other islands) on the planet, but something prevents them from exiting the island ("exiting the island" here is defined as becoming further than a kilometre from the nearest point of the island).
What is the minimum area of the island such that it is able to sustain technology advance?
It's better if you can provide miminum sizes for specific technology levels. I think resource availability is the main problem.
The people in this scenario had inhabited (mostly) the entire planet (like humans do today), but someday they were attacked by an imperialist nation. All (or virtually all) the other nations formed an alliance, but despite that the empire which managed to conquer the world. Most of them (the alliance people) died and the rest (not more than a million) was forcibly moved to the island by the imperials (so they didn't destroy themselves with technology). They didn't have time to bring the technology itself but the scientists managed to arrange a "tech tree".
So just consider they can't exit the island because if they do, the imperial ships will destroy the people which exit. The empire doesn't send people to check directly though, it's part of the story so please don't ask.
The tech tree is shaped like a map (in paper form, of course). There is a book which accompanies the tech tree, containing some information about the technology. Therefore, they don't have to figure out the next step in the technological advance. And yes, safeguarding the tech tree and the book and creating backups are (semi-?)major plot elements.