I agree that having possible examples in mind is a great way to learn mathematics. There are whole books on useful counterexamples, e.g. https://www.amazon.com

I agree that having possible examples in mind is a great way to learn mathematics. There are whole books on useful counterexamples, e.g. https://www.amazon.com/Counterexamples-Analysis-Dover-Books-Mathematics/dp/0486428753 https://www.amazon.com/Counterexamples-Analysis-Dover-Books-... These counterexamples are sometimes a bit involved, but I find they are often useful for understanding the purpose of the technical assumptions that accompany many theorems.