# How to be good at coming up with counter example in Topology

This is a more generalized question, but does anyone have a set of tips or tricks to come up with distinctive examples and counterexamples in Topology and Analysis? More specific, how can people often come up with exotic sequence or mappings between spaces? I can understand the intuition behind some of the simple fractions in the by playing with simple fractions like $$\frac{1}{n}$$, but it seems bizarre to me at this moment how people just come up with maps involving complex numbers, trigonometry between exotic spaces out of nowhere

• There is similarly a book titled Counterexamples in Topology. Also, it might help to think about what properties you are implicitly assuming when you try to come up with examples. E.g., Am I only looking at continuous functions? Differentiable functions? Increasing functions? Compact spaces? Subsets of $\mathbb{R}^n$? Metric spaces? Hausdorff spaces? – kccu Apr 25 at 22:27