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Related: question #879, Most interesting mathematics mistake. But the intent of this question is more pedagogical.

In many branches of mathematics, it seems to me that a good counterexample can be worth just as much as a good theorem or lemma. The only branch where I think this is explicitly recognized in the literature is topology, where for example Munkres is careful to point out and discuss his favorite counterexamples in his book, and *Counterexamples in Topology* is quite famous. The art of coming up with counterexamples, especially minimal counterexamples, is in my mind an important one to cultivate, and perhaps it is not emphasized enough these days.

So: what are your favorite examples of counterexamples that really illuminate something about some aspect of a subject?

Bonus points if the counterexample is minimal in some sense, bonus points if you can make this sense rigorous, and extra bonus points if the counterexample was important enough to impact yours or someone else's research, especially if it was simple enough to present in an undergraduate textbook.

As usual, please limit yourself to **one counterexample per answer.**

3From a pedagogical standpoint, sometimes the minimal counterexample isn't the best one; in particular if it is "too small" to exhibit important general features of what's going on. – benblumsmith – 2011-07-22T12:55:29.060

Someone who knows more about the work being done with the Hasse Principle than I do could probably say something about Selmer's counterexample to an extension of the Hasse-Minkowski theorem to cubic forms. – Ben Linowitz – 2010-03-02T07:04:08.477

@Ben Linowitz. It is in the small and delightful book, J. W. S. Cassels, Lectures on elliptic curves. – Regenbogen – 2010-03-02T14:39:14.490

2@Regenbogen - I am familiar with the proof that Selmer's curve has points everywhere locally but not globally. But that counterexample led many people to study the manner in which the Hasse Prinicple could fail. For example, there is the Brauer-Manin Obstruction. However Skorobogatov has found examples of curves with trivial Manin obstruction and everywhere local points but no global points, so the story is not finished...In my comment I was suggesting that someone more familiar with the current work might use this example. – Ben Linowitz – 2010-03-02T17:23:02.750

1

@Ben Linowitz. Oh I am sorry for saying irrelevant things. I must confess I do not know anything at all. Maybe the following MSRI video might interest you(if you were not already aware of it)... http://www.msri.org/communications/vmath/VMathVideos/VideoInfo/3821/show_video

– Regenbogen – 2010-03-02T17:38:33.617