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1) are notoriously difficult to grasp, both intuitively, conceptually and computationally; 2) are central and critical for furthering one's mathematical understanding; 3) eventually become quite easy ("How is it I could not understand it before?"), because we employ them so much that any mathematician will internalize them after a while.
Setting bar for these three criteria high, four threshold concepts come to my mind:
- basic algebra (it is well-known that many children struggle a lot with middle school maths when transitioning for arithmetics to algebra); - differentiation and integration (AFAIK, differentiation seems more difficult of these two for most students, because it makes them think about graphs in a novel way); - delta-epsilon arguments in real analysis (as 99% undergraduate students in maths can confirm :)); - forcing in advanced set theory (I know nothing about it myself, but I have read several places that it can be a backbreaker; but I am not sure whether it satisfies (3)).
Do you know other examples of such threshold concepts in mathematics?
I would suggest that Adam Roberts, who writes for The Guardian, is very good at finding new interesting SF&F novels: https://www.google.com/amp/s/amp.theguardian.com/books/2023/dec/06/five-of-the-best-science-fiction-and-fantasy-books-of-2023