I was recently intrigued by this style of argument on haskell cafe:

One can write a function
Eq a => ((a -> Bool) -> a) -> [a]
that enumerates the elements of the set. Because we have universal quantification, this list can not be infinite. Which makes sense, topologically: These so-called searchable sets are topologically compact, and the Eq constraint means the space is discrete. Compact subsets of a discrete space are finite.

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I've seen arguments like these "in the wild" during Scott topology construction and in some other weird places (hyperfunctions), but I've never seen a systematic treatment of this.

I'd love to have a reference (papers / textbook preferred) to self learn this stuff!

Thanks

Siddharth