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. Continue reading →