Consider C[0,1] with sup metric.

[coverband]

jjjj

 

[HallsofIvy]

Okay, I'm considering it!

(How long? I have a class to go to soon!)

For those who are wondering, "C[0, 1]" is the set of functions, f(x), that are continuous on the interval [0,1].

The "sup metric", also called "supremum metric" measures the "distance" between functions by the largest difference between values: max |f(x)- g(x)| over all x between 0 and 1. Notice that since f and g are continuous so is f- g and so the maximum over the closed and bounded interval [0, 1] does exist.

Of course, one would hope that there is some "problem" associated with this.