It is not unusual that a single example or a very few shape an entire mathematical discipline. Can you give examples for such examples?
I’d love to learn about further basic or central examples and I think such examples serve as good invitations to various areas.
I asked this question over mathoverflow and it yielded around 100 examples. They are not equally fundamental and they are not equally suitable to be regarded as “examples,” but overall it is a very good list. If you see some important example missing please, please add it. Here are the examples classified to areas. (Of course, sometimes, the same example may fit several areas.)
Logic and foundations:
(~1890), Russell’s paradox (1901), Halting problem (1936), Goedel constructible universe L (1938), McKinsey formula in modal logic (~1941), 3SAT (*1970), The theory of Algebraically closed fields (ACF) (?),
Real and Complex Analysis: