A simple example of this distinction is the following: one can define a function on the surface of the earth, whose value at a point is the current temperature at that point. One could similarly define such a function on the entire universe, sending a location to its temperature. Since the earth is compact, it follows that there is some point on earth which is the hottest at any given time. However, if the universe is unbounded, it is not compact, and we can not reach the same conclusion! There may be points in the universe which have arbitrarily large temperatures, so that the function has no maximum. | A simple example of this distinction is the following: one can define a function on the surface of the earth, whose value at a point is the current temperature at that point. One could similarly define such a function on the entire universe, sending a location to its temperature. Since the earth is compact, it follows that there is some point on earth which is the hottest at any given time. However, if the universe is unbounded, it is not compact, and we can not reach the same conclusion! There may be points in the universe which have arbitrarily large temperatures, so that the function has no maximum. |