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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

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For counterexamples, just think: "mission sabotage". In other words, deliberately try to break a given statement.

There are generally no "tips", "tricks", "recipes", or anything else of a universal caliber. (When there are, they are so valued that you will surely run across them.) Mathematics is an art as much as it is a science: one tries, examines for errors, and corrects if needed, as many times as it takes.

The best there is in the direction you are asking is learning a sufficiently rich arsenal of counterexamples. To help with that, Olmsted and Gelbaum have written Counterexamples in Analysis, which is a great and highly beneficial read.

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    $\begingroup$ 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? $\endgroup$ – kccu Apr 25 at 22:27
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    $\begingroup$ Thank you very much for the comment. This seems like an excellent book for me to start with. It is just I started taking my first algebraic topology course this semester, but I feel dull as every time I think about some possible theorem to prove, the stack-exchange community would just come up with bizarre (at least to me) counter-example in a short period that would take forever for me to verify. $\endgroup$ – Joe Martin Apr 25 at 23:35
  • $\begingroup$ @Joe Martin, that's because they say it (or a similar counterexample) before. Don't judge your ability to conceive of brand new ideas against the experience of the entire MSE community. $\endgroup$ – Mark S. Apr 26 at 11:20
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I think that it would indeed be odd for people to come up with exotic counterexamples to innocuous conjectures out of nowhere, as you say. Really, what is guiding those counterexamples is a lot of time and experience spent with problems and the material. When you are reading the statement of a theorem, try seeing what happens when you omit a hypothesis to see what may go wrong, and talk to people about it, either here online or in person to share your thoughts. The more you learn, the more connections you will make, and eventually you will begin to see more as you synthesize that knowledge.

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