I had an analysis exam today, and the last question was: Give an example of a set that is pointwise compact, (uniformly) equicontinuous, and NOT totally bounded.
I know has to be open, otherwise by the Arzela-Ascoli theorem, it would be compact and therefore totally bounded, but unfortunately, I got to this question with ~3 minutes left, and was only able to come up with . I am pretty sure this doesn't work, so what would an example of a function with the above properties be?
Note that is the space of all bounded functions from the reals to the reals.