Mission Counterexample!

An open ball inside the closed ball and isolated from its boundary.

Providing a counterexample can be the most challenging and frustrating exercise for me.  For example, one of my recent homework exercises was to give an example of a metric space where the closure of an open ball, , could differ from the closed ball, .  This is counter to naive intuition and experience: the closure of an open ball in is the closed ball.  It was actually not difficult to find a solution to the above exercise – can you find one?  However, I am not content with finding one.  If I can find one counterexample, then there is likely to be plenty of counterexamples.

It is a source of pride for me if I can list two or three counterexamples when the exercise calls for only one.  Usually the constraint of time limits me to just one counterexample.

Generally, I sketch what the counterexample must look like.  For the above, the closure of the open ball is a proper subset of the closed ball.  Therefore, the elements in the boundary of  the closed ball must be isolated from the elements in the open ball.  Sometimes, this approach is immediately fruitful; when it fails to be, it can be frustrating.  With the glut of information available on the internet, it is very tempting to find a counterexample there.  Normally, I give myself a week after I turn in my homework.  If I can’t think of a counterexample by the end of the week,  I’ll search for one on the internet.I’ll leave you with another exercise: find a sequence which converges to zero but is not in any -space for .  This is a little easier if you have the Cauchy condensation test in mind.  I would like to hear some of your favorite counterexamples.